MATHEMATICS 


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result in dismissal from the University. 


UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 


JULIE 49 


AUG 13 1974 


L161—O-1096 


SS OF, 


 ARIPHMETIO 


AND 


BY JAMES H. PORTER, A. M., 


TEACHER OF MATHEMATICS AND NATURAL PHILOSOPHY. 


FOURTH EDITION. 


| NEW-YORK: 

PI BROY AN D RE ED, PRINTERS, 
Ores. Noe Se Spruce Street. 

aes 1845. 


I ae RPER, 1 


wv the Southern District of the State of N ew-York 


BILE, A * 
Sab 3 oa ye oe e yh 
RPTIBTAREAIE LO’ AIDA Ge 


PREFACE. 


Tuer science of numbers is universally conceded to be 

an important one to every class in the community. In its 

, cultivation and attainment a// are deeply interested. The 
“ *merchant, the mechanic, the professional man, the manu- 
facturer, agriculturist, and laborer also require, in a 
‘ greater or less degree, an’acquaintance with its principles, 
*° ona correct understanding of which depend their pros- 
x perity in the prosecution of their several pursuits. The 
Re intricacies that have entered into this branch of necessary 
knowledge, have deterred many from undertaking its 
successful acquisition. The labor, the time, the study 
Ss required to achieve a complete mastery over numerical 
“ science, interpose | formidable objections to the system 

2 ‘that has been. so Jong established ; which, although it 
a? > cornannds respect for its age, yet its numberless defects 
* furnish often almost insurmountable difficulties to its easy 
~ and rapid attainment.. 

To reduce the science of numbers toa greater degree 
‘of simplicity and facility of acquisition; to introduce into 
= "the plan of antiquated theories an easier, a surer, and 

~ quicker method of computation, is an important desidera- 


be | 


iv PREFACE. 


- 


tum, one that necessarily interests the actors in eyery 
department of business, and recommends itself to the 
fayorable consideration of the community. 

‘Many, who have in some measure thrown off the 
shackles of prejudice, and have endeayored to become 
reformers in this interesting field of enterprize, have failed 
substantially in accomplishing any very important results. 
They who-would be improvers in oral, mental, and prac- 
tical arithmetic and mathematics, have not as yet been 
able, with all their investigation, to present to the world 
a plan, by means of which these branches of science may 
be taught by a simplified method, and communicated 
with facility and precision to the understandings of the 
learners. 

To remedy these defects, and to supply the means for 
facilitating the acquisition of arithmetical and mathematical 
knowledge, has engaged much of the time and attention 
of the author of this work : the persevering efforts made 
have been crowned with complete success ; and the author 
begs leave to tender his congratulations to the public on 
the important improvement thereby made to the cause of 
arithmetical and mathematical science, whereby its acqui- 
sition is rendered more easy and more certain, requires 
much less time in its practical operations than heretofore, 
and reduces it to so simple.a system, that even the most 
obtuse understanding can readily comprehend its princi- 
ples. The saving of time is always an important item in 
the calculation of business; and the more rapidly correct 
computative results can be arrived at, the better for those 

who have embarked in their transaction. © 
‘This important improvement in the science of numbers, 
by which much time is saved and greater certainty ob- 


s . 


PREFACE. V 


tained, is now about to be introduced to the public, for 
their approbation and accéptance. | 

By way of exposition, it is necessary to state, that the 
whole of the principles of the science generally are con- 
tained and expressed i in three words, viz: — 


IcREasn. Decrease. EQuautrry. 


“Increase and decrease comprise addition, subtraction, 
multiplication, and division. _ Equality is the answer 
required, fixed by the question propounded under two 
different names, after which: it is represented or expressed 
by a statement or equation, reduced by increase and . 
decrease to the lowest term, equality, or the answer 
required. The science thus far concentrated, nothing 
remains but to simplify expressions or statements, to 
facilitate increase and decrease, which constitute the 
purport of the present publication. 

It is expected that the fundamental principles of the 
science will be taught by gradual and persevering prac- 
tice; it will liberate the minds of the teachers from the 
imperious necessity of propounding useless and irrelevant 
questions, and conduct the understanding along the ave- 
nues of knowledge, until the principles in which the 
science is based are fully understood. A progressive and 
reasonable instruction will enable teachers to make pupils 
of every grade in society perfectly acquainted, at the age 
of twelve years, with the science generally, adapted com- 
pletely to the attainment and prosecution of mercantile, 
mechanical, and mathematical knowledge. 

For a number of years it was the constant study of the 
author to bring into-organic operation his simplified plan 
of instruction, as regulated by his fundamental principles. 


vi PREFACE, 

This method of calculation affords the power of per- 
forming calculations in whole numbers, even when the 
question is composed of whole fractions, or number and 
fractions. By an easy process in the statement the frac- 
tions are rejected, the solution or calculation is performed 
by the pure proportion of all variation of measure, weight, 
money, &c., of the whole world, entirely by whole num- 
bers and in an uninterrupted series. It teaches to obtain, 
by a succession of pure proportion, an answer to any 
arithmetical proportional question proposed. ‘The rule of 
three, or the rule of proportion, named also the “ golden 
rule,’ has not this power. By this rule. we are often 
compelled to make four, five, and more statements before 
we are able to obtain the answer required. These pro- 
ceedings, by the common rule of caleulation with frac- 
tions, render the. process circumstantial and confused to 
the scholar, and difficult to impress on his memory ; but 
the rule of pure proportion teaches, in an easy, agreeable, 
and unavoidable manner, all the rules in general, as rule 
of three, tare, barter, fellowship, interest, reduction, loss 
and gain, exchange, and others ; and even in the solution 
and statements of these questions, wherein it is now 
necessary to employ several of these rules, the rule of 
pure proportion will suffice ; and it also performs the cal- 
culation always without interruption, and in whole num- 
bers. By this rule all circumstantial calculation of 
fractional numbers are avoided, and, by the shortness in 
whole numbers, more agreeable too, than the circumstan- 
tial calculation with compound numbers ; and it may be 
said, without hesitation, that the rule of pure proportion 
affords, in all business of common life, the same easiness 
as the decimal system does in the science of mathematics. 


SS 


PREFACE, Vil 


To enlarge this work by a long preface is not the inten- 
tion of its author. It may speak for itself. It will be 
found, on examination, to do what it professes, viz., to 
teach an easy method of calculation, and to afford inter. 
esting and necessary knowledge to all men of business. 

The pupil, even when he walks out for recreation, will 
find: a subject for his thoughts and an agreeable little 
companion in this work. The amusing variation will 
afford to the scholar principles which will enable and 
~ animate him to perform questions hitherto unknown in 

any system of arithmetic; by the knowledge of pure pro- 
portion and true judgment, which this system of figures 
gives of fractions, the young pupil becomes, in the course 
of his studies, better prepared for the higher branches of 
mathematics, and the tutors will not have half the trouble 
to ingraft durable principles of calculation on his memory. 
_ Finally, it may be observed, that the author of this 
method of calculation has shown a fixed rule, that will 
not be found in any system of arithmetic—a rule to find 
the pure proportion of all things. Besides, he has adjusted 
the necessary pure proportions in a few pages at the end 
of the work, and placed there also a few sheets of writing 
paper, for the purpose that new pure proportions, desired 
and found after this rule, may be neatly traced thereon. 


Tue AUTHOR. 


Fovagrmkio a eal news higag od 


(ceasdoteid ‘weds oot. aah, Seceamet s@ion: ind 
elegans: eit Mgt vied thes ir ena a : 


 PORTER’S NEW SYSTEM 


OF 


ARITHMETIC 


AND 


MATHEMATICS. 


- 


ARITHMETIC, 


Aritumetic is the art or science of computing by num- 
bers, and consists both in Theory and Practice. The 
Theory considers the nature and quality of numbers, and 
demonstrates the reason of practical. operations. The 
Practice is that which shows the method of working by 
nuimbers, so as to be most useful and expeditious for 
business, and is comprised under five principal or funda- 
mental rules, viz., Notatcon or Numeration, Addition, Sub- 
traciion, Multiplication, and Division ; the knowledge of 
which-is so necessary, that scarcely anything in life, and 
nothing in trade, can be done without it. 


fp mcr 


NUMERATION 


Se 
TEACHETH to express any proposed number by these ten 
characters: 0, 1, 2, 3, 4,'5, 6, 7, 8, 9—0 is called a cipher, 
and the rest figures, or digits; the relative value of which 
depends upon the place tiey stand in when joined to- 
gether, beginning at the right hand, as in the following : 
ox 


10 NUMERATION. 


TABLE. 


o Tens of Millions. 
o Hundreds of Thousands. 


co Hundreds of Millions. 
on. Tens of Thousands. 


«x Millions. 

= Thousands. 
eo Hundreds. 
wo Tens. 

- Units. 


Though the table consists of only nine places, yet it 
may be ” extended to more places at pleasure; as, after 
hundreds of millions, read thousands of millions, ten 
thousands of millions, hundred thousands of millions, bil- 
lions, trillions, quadrillions, quintillions, sextillions, septil- 
lions, octillions, nonillions, decillions, undecillions, &c., as 
in the following example: 


Quadrillions. Trillions. Billions. Millions. Units. 


eer Pee [Px Po C/N 
867 890 707 928° 679° 437 963.897 »- 234°278 
eer Ney ere me ee ee ee ee Ss 


To write down numbers. 


Rule. Write down the figures as their values are express- 
_ ed, and supply any deficiency in the order with ciphers. 


EXAMPLE. 


Write down the following numbers in order: 
Twenty-nine. F " 
Two hundred and forty-six. 

Six thousand nine hundred and one. 
Eighty-four thousand three hundred and nine. 
Six millions two hundred and sixty- eight. 
Highty-nine millions and ninety. 

Four millions four hundred thousand. 

Nine hundred and nine millions. | 

Seventy millions seventy thousand and seventy. 
Lwelve hundred and forty-six millions. 


SIMPLE. ADDITION. 11 


Eight hundred millions forty-four thousand. 

Two thousand five hundred and forty-three millions, 
four hundred and thirty-one thousand. 

Sixty-nine hundred, nine thousand and seventy-five. 


SIMPLE ADDITION 


Tracnetn to collect numbers of the same "denomination 
into one sum. 


Rule. Place the numbers under each other, so that units 
may stand under units, tens under tens, and ‘so. on, and 
draw a line under them. Add the first row, or right- 
hand column, and find how many tens are contained i in 
them; set down the remainder, and carry as many units 
or ones to the next column, as there are tens. In like 
manner, carry the tens of each column, till the whole be 


finished. 


Proof. Begin at the top of the sum and reckon the 
figures downward, in the same manner as they were 
added upward ; and, if it be right, this aggregate will be 
equal to the first: or, cut off the upper line of figures, and 
find the amount of the rest; then, if the amount and up- 
per line, when added, be equal to the sum total, the work 
is right. 


EXAMPLES. 
(1.) (2.) (3.) 
4 2,2 2345 
— 3 3 2345 
7 44 4,3. 5.2 
9 5.5 5 432 
3 6 6 6625 
; 6 oe — 
-—— g2 6 21°0 99 
29 am 
25 


SIMPLE ADDITION, 


12 


(5.) 


~~ 
. 


tak 


Neer OM 
Kent © 
DoOrnrAstAN 
HOOD 
montownn 
Cyr CO 1 N SH 
OO OIWIAN 
Horowmo 
QODMANr OP 
~ NH rH HN 
Hop wt OO 
co 67 SO SOD 


<tcO OD ON 
mm0o ow 


Vraonnrnd 


ONMOAA 


Add the following sums : 


(6.) 3784, 94, 96484, 04, 978649, 708. 


AIHW OAOAMUADODOYrO 
MOMFrFANraATHNANHOD 
HHOWMMOrOMANAQDTHR WON 
DOMNRSHDONNDYMOND 
RDOPrROANMNRMWOAGANON HOM 
SCHHHODHHORNOCAMDYWHO 
Vat eHKCNANQHTDOKrAMNMOUOMND + 
AMAMANMNDNORAWMOWOOD 
OttOrnatAtONoerom 
WOAKrMOFrrANAO AMIN WL 


Ans. $240, 


11. A man borrowed of his neighbor 30 dollars at one 
time, 106 at another, 67 at another, and 37 at another: 


how much did he borrow in the whole ? 


SIMPLE ADDITION. 13 


12. Four boys collected Chestnuts; John had 4096, 
Peter had 16784, Charles had 11590, and David 557; 
how many were there in the whole ? Ans. 33027. 

13. Four boys, on counting their apples, found Andrew 
had 67, Bennet 11 more than Andrew, Charles had 101, 
and Daniel had 16 more than Charles ; how many apples 
had they all ? 

14, The deluge happened 2348 years before the birth 
of our Saviour, and America was discovered 1492 years 
after it; how many years intervened ? 

15. A farmer raised, in one year, 60 bushels of oats, 
940 bushels of wheat, 370 of corn, and 80 bushels of pota- 
toes; how many bushels did he raise in all? 

16. A gentleman has six debtors, A, B, C, D, E, and. 
F: A owes him 500 dollars; B owes him as much as A, 
and 90 dollarsmore; C owes him as much as A and B both; 
D owes him 67 dollars; E owes him as much as D and 
A’s debts amount to; and F owes him 64 dollars more 
than the sum of A, B, C, D, and E’s debts added together. 
What is the whole amount due him 1? 

17. Four thousand two hundred and sixty-nine is one- 
sixth of some number; what is that number ? 

18. The ship Mary has just arrived from London, and 
one-fourth of her cargo is worth six thousand eight hun- 
dred and four dollars; what is the whole cargo worth ? 

19. Sir Isaac Newton was born in the year 1642, and 
died in the eighty-fifth of his age ; in what year did he die ? 

20. If four bushels of wheat make one barrel of flour ; 
and the price of wheat be one dollar per bushel; what 
will 400 barrels of flour cost ? | 

21. King Charles the martyr was beheaded in the year 
1648, and 130 years have i since that per iod ; what 
year is it now ? 

22. From the creation of the world to the flood was 
1656 years; from thence to the building of Solomon’s 
temple was 1336 years; thence to the birth of our Saviour 
was 1008 years; in what year was the birth of Christ ¢ 

Ans. 4000. 
236A gentleman owns one-eighth of a bank, and his 
part is worth 26,000 dollars; what is the value of the 
whole bank ?. Ans. $208,000. 


14 SIMPLE SUBTR ACTION. 


24, The ship Lion sailed from Boston, bound to the 
port of Liverpool, with a cargo consisting of merchandize 
and specie; the value of the merchandize was 4444 dol- 
lars; the value of the merchandize was one-fourth of the 
value of the specie, and the value of the specie was one- 
half of the value of the ship; what was the value of the 
ship and cargo ? Ans. $57,772. 

25. If the cargo of a ship is. worth 14,678 dollars, and 
the value of the cargo only one-sixth of the ship; what is 
the value of the ship and cargo ? Ans. $102,746. 

26. Suppose the distance from A to B is 44 miles, and 
the distance from A to B is one-half the distance from b 
to C, and the distance from B to C is one-third of the dis- 
tance from C to D, and the distance from C€ to D 1s one- 
fourth of the distance from D to E, and the distance from 
D to E is one-sixth of the distance from E to F, and the 
distance from E to F is one-eighth of the distance from F 
toG; require the distance from AtoG? 

Ans. 58,476 miles. 


SIMPLE SUBTRACTION 


TEAcHETH to take a less number from a greater of the 
same denomination, and thereby show the difference. 

The greater is called the minuend, and the less the 
subtrahend. 

Rule. Place the subtrahend, or less number, under the 
minuend or greater, and subtract units from units, tens 
from tens, and so on; if any figure of the subtrahend be 
greater than the corresponding one of the minuend, add 
ten to the upper figure, and then subtract the lower from 
the sum, set down the remainder, and carry one to the 
next figure of the subtrahend. | 

Proof. Add the remainder to the subtrahend, and if the 
sum is equal to the minuend, the work is right. 


EXAMPLES. 
2. 


— 


648 
438 


( 
89358 
88356 


SIMPLE SUBTRACTION 15 


(3.) [dein cs 
Foi oO 4o30456.778 SE DE IE Be ie Oe: 
Bid tad (Oe | Ord 609..7..6.4 4°6.8 7 
bud46.678 2 6r ln Stor 1 .,.95,4.9'7:.855 :428-6 

(5.) (6.) 

G6p4i Gully 9. 3°8 36.7 5-070 0 Biata.d int 
4556, Gved Bb pS Au Fa Ge8 A Ae (4 Oe oe oO 
(7, (S.) 

4 $8: .68 7% 3:-S:c6. 044-97 3 0a 0s HART G8 A 689: 3 
47 9 8.64.53 5 9 02070 0:0 

(9. big, (0%) 

4°36 Br b2O64s 648.7 200000000 

3000000186 19 9:99 9:9 9-9 
1. (12.) 

J: S161 4+ 337-0 0010.0 8x64. Be 8r6 4638377453 

1 Sass 32: Aes 208537 


es es 


13. A man, having 478 dollars, lost 149 dollars in gam- 
bling; how many had he left 2 Ans. $329. 
14, A gentleman bought a wagon for 108 dollars, and a 
harness for 42 Abilis" what did the wagon cost him 
more than the harness ? Ans. $66. 
15. Ifa man have 2476 dollars, and he spend 675 dol- 
lars of it; how much will he have left ? Ans, $1801. 
16. Aman bought a chaise for 214 dollars, and, to pay 
for it, gave a gig worth 47 dollars, and the rest in money ; 
how much money did he pay? Ans. $167. 
17. America was discovered by Christopher Columbus 
in 1492; how many years had elapsed when _ hostilities 
commenced in the revolutionary war, 17751 Ans. 283 years. 
18. General George Washington was born in 1732, and 
died in 1799; what was his age ? Ans. 67 years. 


16 SIMPLE MULTIPLICATION. 


19. Sir Isaac Newton was born in 1642, and died in 1727; 
what was his age at the time of his death? Ans. 85 years. 
20. A man in the year 1820 was 67 years of age; in 
what year was he born 4 Ans. 1753. 
21. What is the difference between twice twenty-seven 
and three times forty-five ? Ans. 81. 
22. How much is 1200 greater than 365 and 721 added 
together ? Ans. 114. 
23, From New London to Philadelphia is 240 miles, 
Now, if aman should travel five days, from New London 
toward Philadelphia, at the rate of 39 miles each day, how 
far would he then be from Philadelphia ? Ans. 45 miles. 
24, What other number with these four, viz., 21, 32, 16, 
and 12, will make 100? Ans. 19. 
25. A wine merchant bought 721 pipes of wine for 
90,846 dollars, and sold 543 pipes thereof for 89,049 dol- 
lars; how many pipes has he remaining or unsold, and 
what do they stand himin? Ans. 178 pipes, and $1797. 
26. Joe Careless received prize-money to the amount 
of 1000 dollars, after which he lays out 411 dollars 41 cents 
for a span of fine horses, and 123. dollars 40 cents for a 
gold watch and a suit of new clothes, besides 359 dollars 
and 50 cents he lost in gambling; how much will he have 
left after paying his landlord’s bill, which amounts to 85 
dollars and 11 cents ? : Ans. $20 58 ctss 


4 


SIMPLE MULTIPLICATION 


Is a compendious way of adding numbers of the same 
name. ., 7 

The number to be multiplied is called the multiplicand. 

The number which multiplies is called the multiplier. 

The number arising from:ihe operation is called the 
product. 

Rule. Place the multiplier under the multiplicand, and 
multiply the latter successively by the significant figures of 
the former; ifthe multiplier consists of more figures. than 
one, place the right hand figure of each product under the 
figure from which it arises; then add the several pro- 
ducts, and their sum is the product of both factors, and the 
answer required. 


aA AEE LALEPALA ALLELE EE ES 


WNW DH NHONNWNWNHNYNHNNNNNNNHKHYWND 


MULTIPLICATION TABLE. 


1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 


OMDWIOMIP WWE 


SIMPLE MULTIPLICATION. 


. are 
és 


2 


WWwWWHWWwWeUWwwwwwuNnwuwnNwowww 


AMMANMAAAAAAAMIAaAananna4nan 


times 
66 


éé 


are 


17 


_ 
@ 


6 


DADDDDDDANDDADDWNDNWDWDANAMD 


SIMPLE MULTIPLICATION. 


are 


¢é 


7 


of 


9 
) 
9 
4 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
5 
9 
9 


7 
7 
7 
7 
7 
7 
7 - 
7 
7 
7 
7 
7 
7 
7 


7 
7. 
3 
7 


~ times 


ce 


SIMPLE MULTIPLICATION, 


are 
“é 


times 


Po pet ee pe RE RR 
OMNVONBRWNHWDEFOUWUOANAUOPWD 


* 


SIMPLE MULTIPLICATION. 


15 times 


6e 


ee ce ee we Od ce 
SODMDVHNMEWNHWHFPOKOANAAFWNHR 


WOMANAnAKWDHORF 


SIMPLE MULTIPLICATION. 


(1.) 


18 times/1 are 18 19 
18 bs 2 te 36 19 
18 ve 3 - 54 19 
184582 Asai’ 72 19 
18 5 SS 5 ES 90 19 
18... GirsackS ts 1.08 19 
18.40% Y aakie. 2126 19 
18 ¢¢ 8 ey cheba 19 
18. wuss Din HE Dae 19 
18 f 10 ‘f 180 19 
18 ih 1. Ks 198 19 
18 a 12 fs 216 19 
18 3 13 66 234 19 
18 sf 14 a 202 19 
18 iF 15 6 ig 270 19 
18 rf 16 ¥ 288 19 
18 Ki 17 £41 306 19 
18 He 18 kf 324 19 
18 é 19 46 342 19 
18 § 20 £6 360 19 
20 times 1 are 20 20 
20 . 2 - 40 20 
20 SERerhIgh iver ogy 20 
20 Fe 4 se 80 20 
20° cs 5) et 100 20 
20 -* 6 - 120 20 
20 sin 7 vd 140 20 
20 & 8 th 160 20 
20 ee 9 Rk 180 20 
20 As 79% O36 20 
EXAMPLES. 


bicBy ity Ay? Balke ni 


2 


eee Gere en Genensonetoy ae 


86824686 

(3.) 
64784656 
4 


eS MESSER een 


times 4 are 17 
iG 2 6c 38 
66 3 éc 57 
6c 4. 73 76 
Silat Gia —-O5 
fds. iGO, 114 
ho ania bealas 
66 \ 8 6é 152 
€ OF eehhor A Bh 
“4 1Qac fi £90 
(73 11 6é 909 
sf in HS. 0228 
66 13 $6. DAM. 
Se shie 14 ‘6° 966 
66 15 66 985 
éf 16 és. 1304 
‘cc We “392 
f A8s cf fo3 B42 
6 19 ‘ce 361 
‘oi Ore 246 380 
times 11 are 220 
TATE Sy kee aA 
i Pe ee Poet 
66: 14 6é 980 
- 15 se” 300 
me 16 rer oLO 
“af a “340 
Bf AS 286860 
ee LD F386 
SPM EE MOD) 
(2.) 
35683 75 
3 
0705125 
Soir otf4d . 
86964679 7 


26 


22 SIMPLE DIVISION. 


(5) | (6.) 
69 743.9 7'9 4 v.50 (6. Fo ee 
B 408 Ao Otee 


7. What is the value of 29 pairs of men’s shoes, at 1 


dollar 51 cents per pair? Ans. $43,79 cts. 
8. What cost 140 reams of paper, at 2 dollars 35 cents 
per ream ? Ans. $329. 


9. In one pipe of wine there are 126 gallons; how 
many gallons are there in 15 pipes ? Ans. 1890 galls. 
10. A merchant bought at one time 278 barrels of fish : 
- at another time three times as many, wanting 184 barrels, 
at 9 dollars per barrel ; what did he pay for the whole ? 
* Ans. $8352. 
11. A man bought 675 sheep, at 2 dollars 25 cents per 
head ; also, 46 pigs, at 2 dollars per head; what did he 
pay for the whole? Ans. 3 1610,75 cés. 
12. A gentleman bought 2428 barrels of flour at one 
time ; at another time as many as at the first, wanting 987 
barrels ; he lost as many as he bought at the last time, 
wanting 764 barrels; sold the remainder at 8 dollars per 
barrel; what did they come to? Ans. $25536. 
13. What will,one year’s board come to, at 6 dollars 
per week ? Ans. $312, 
14. What will 40,000 brick cost, at 3. dollars per thou- 
sand 1% Ans, $120,000. 


TA a TT 
SIMPLE DIVISION 


TEACHETH to find how often one number is contained in 
another of the same name.’ 

The number to be divided is called the dividend. 

The number by which to divide is called the divisor. 

The number of times the divisor is contained in the di- 
vidend is called the quotient. 

The remainder, if there be any, will always be less than 
the divisor. 

Rule. Qn the right and left of the dividend draw a 
curved line, and Write the divisor on the left hand, and 
the quotient, as it arises, on the right. Find how many 


SIMPLE DIVISION. 23 


times the divisor is contained in as many figures as are 
necessary in the dividend, and write the number in the 
quotient. 

Multiply the divisor by the quotient figure, and set the 
product under that part of the dividend used. 

Subtract the product last found from that part of the di- 
vidend under whichit is placed, and to the right hand of 
the remainder bring down the next figure of the dividend ; 
divide this number as before, and so on, till the whole is 
finished. If it be necessary to bring down more figures 
than one to the remainder, in order to make it as large as 
the divisor, a cipher must be written in the quotient for 
every figure so brought down, till the number be suffi- 
cient to contain the divisor. 

Proof. Multiply the quotient by the divisor, and to the 
product add the remainder, and the sum will be equal to 
the dividend, if the work is right. When there are ciphers 
annexed to the divisor, cut off the ciphers-from it, and the 
same number of digits from the dividend, then divide the 
remaining figures by each other, as usual, the quotient is 
the answer ; and what remains, placed before the figures 
cut off, is the true remainder. 

When the divisor does not exceed twelve, or is a com- 
posit number, or when ciphers may be cut off from it, the 
division may be shortened by multiplying and dividing 
mentally, and writing the quotient under the dividend. 


: ‘. EXAMPLES, 
5 A) (2.) 
4)568743864 6)6894868 
142185966 1149144—4 rem. 
(3-) (4) 
9)9763856 _-12)8764864 


lena areas ee a en 


1084872—8 730405—4 


24 


* 


5.) 
4)87643(21910 
4 


(7.) 


14)6898(492 


56 

129 

126 
38 
28 


10 


(9.) 
40)7364(184 
40 


336 

320 
164 
160 


4 


stMPLE DIVISION. 


(6-) 


17 
16 
16 
16 
43 
40 


——s 


3 


36) 


*)97643(12205 
8 


remainder. 


(8:) 
87436(2428 
rk : 


(10.) 


360)978964(2302 


720 
2589 
2580 
964 
720 


es 


244 


SIMPLE DIVISION. 25 


ae eae WES. 12.) 


(E2.) 
3956)34585539543 (8742552 — 6,00)74389486,00(12398247 
6 


31648 


29375 
27692 
16833 
15824 
10099 
7912 
21875. 
19780 
20954 
19780 
11743 
7912 


eer ere ee 


3831 


14 


Be 


os 


23 
18 
58 
54 
49 
48 
14 
12 
28 3 
24 
46 
42, 


4 


13. If the sum of 262,200 dollars were equally divided 
among 345 men, how many dollars would each receive ? 


Ans. $750. 


14. If 333,046 dollars were equally divided among 456 
men, how tack would each receive ? Ans. $540. 
15. Sold 345 bushels of wheat for 2415 dollars ; what is 


it per bushel ? 


Ans. $7, 


16. Sold a farm, containing 365 acres of land, for 8395 
dollars; how rads was it per acre? Ans. $23. . 
17. If a prize, worth 36,9000 dollars, be equally divided 
among 450, how much would each man receive ? 


Ans. $82. 


18. Bought a piece of cloth for 363 dollars, at 3 dollars 
per yard ; how many yards in the said piece of cloth ? 


Ans. 121, 


26 SIMPLE DIVISION. 


19. A man haying 5520 bushels of corn, wishes to cis it 
into bins, each holding 16 bushels ; how many bins will 
it take? Ans. 345 bins, 

20. A regiment of soldiers, consisting of 500 men, are 
allowed 1000 pounds of pork per day ; how much is each 
man’s part % Ans. 2 lbs. 

21. Write down 4617, multiply it by 12, divide the pro- 
duct by 9, and add 365.to the quotient, then from that 
sum subtract 5521, and the remainder will be just one 
thousand. Try it and see. 


TABLE OF MONEY, WEIGHTS, MEASURES, cc. 
1. Federal Money. 


10 mills (m.) make 1 cent, marked é. 

10 cents, 1 dime, d. 

10 dimes, , 1 dollar, $. 

10 dollars, 1 eagle, E. 
2. Sterling Money. 

4 farthings make 1 penny, d. 
12 pence, ! shilling, gs. 
20 shillings, 1 pound, on 

3. Froy Weight. 
24 grains (g7.) make 1 pennyweight, marked dwt. 
20 pennyweights, 1 ounce, Oz. 
12 ounces, . 1 pound, lb, 
4. Avoirdupois Weight. 
16 drachms (d7.) 1 ounce, OZ. 
16 ounces, ON pound, lb. 
28 pounds, 1 quarter of a hundred weight, qr. 

4 quarters, 1 hundred weight, cut. 

20 hundred weight, iton. 7" | x 


By this weight are weighed all coarse and drossy goods, 
grocery wares, and all metals except gold and silver. 


5. Apothecaries Weight. 


20 grains (g7.) make 1 scruple, iS) 
3 scruples, 1 dram, 3 
8. drachms, 1 ounce, 3 

12 ounces, - 1] pound, ib 


Apothecaries use this weight in compounding their 
medicines, 


~ 


SIMPLE DIVISION. Qf 


¢ 6. Cloth Measure. 


4 nails (ma.) make 1 quarter of a yard, gr. 

4 quarters, 1 yard, * yd. 

3 quarters, - 1 Ell Flemish, Et. Fl. 
5 quarters, ~ 1 Ell English, Hh. EE. 

6 quarters, 1 Ell French, Ei. Er. 

7. Dry Measure. 

2 pints, ( pt.) make 1 quart, | qt. 
8 quarts, 1 peck, phe 
4 pecks, 1 bushel, bu. 


This measure is applied to grain, Beans: flax-seed, salt, 
oats, oysters, coal, &c. 


8. Wine Measure. 


4 gills (g7.) make 1 pint, a) gs 

2 pints, 1 quatt, gt. 

4 quarts, —— 1 gallon, gal, 
314 gallons, ~~ 1 barrel, bl. 
42 gallons, 1 tierce, . tier. 
63 gallons, 1 hogshead, hhd. 

2 hogsheads, ‘1 pipe, D. 

2 pipes, 1 tun, at 


All brandies, spirits, mead, vinegar, oil, &c., are mea- 
sured by wine measure. Note. 231 solid inches make a 
gallon.. 


_ 9. Long Measure. 


3 barley corns (d. c.) make 1 inch, marked ine 
12 inches, 1 foot, rs i 

3. feet, 1 yard, yd. 

54 yards, 1 rod, pole, or perch, «rd. 
40 rods, 1 furlong, fur. 

8 furlongs, 1 mile, M. 

3 miles, 1 league, * lea. 
694 statute miles. 1 degree, on the earth, 


360 degrees, the circumference of the earth. 


The use of long measure is to measure the distance of 


places, or any other thing where length is considered with- 
out regard to breadth. 


N. B. In measuring the height of horses, 4 inches make 
1 hand. In measuring depths, 6 feet make 1 fathom or 
French toise. Distances are measured by a chain, four 
rods long, containing one hundred links, 


28 SIMPLE DIVISION. 


10. Land, or Square Measure. ~ 


114 square inches make 1 square foot. 
J square feet make 1 square yard. 
304 square yards, or t 


2724 square feet, 1 square rod. 


40 square rods, - 1 square rood. 
4 square roods, 1 square acre. 
640 square acies, 1 square mile. 
11. Solid, or Cubic Measure. 
1728 solid inches make 1 solid foot. 


40 feet of round timber, or 
50 feet of hewn timber, 
128 solid feet, or 8 feet long, 
4 wide, and 4 high, t 
All solids, or things that have length, breadth, and depth, 
are measured by this measure. N.B: The wine gallon 


contains 231 solid or cubic inches, and the beer gallon 
282, <A bushel contains 2150,42 solid inches. 


1 ton or load. 


1 cord of wood 


12. Time. 
60 seconds (S.) make 1 minute, marked M. 
60 minutes, 1 hour, h. 
24 hours, 1 day, d. 
7 days, 1 week, w. 
4 weeks, 1 month, mo. 
~13 months, 1 day and 6 hours, 1 Julian year, yf. 


Thirty days hath September, April, June and Novem: 
ber; February twenty-eight alone, all the rest have thirty- 


* il 
i “one. 


N.B. In Bissextile, or leap year, February hath 29 days. 
13. Circular Motion. 


60 seconds (”’) make 1 minute, oH 
60 minutes, : 1 degree, ° 
30 degrees, 1 sign, é Ss. 


12 signs, or 360 degrees, the whole great circle of the 
Zodiac. 


29 


AND MEASURES. 


WEIGHTS 


6 \I 

GSP mal 
POOTL 9661 
IESCSE FOc6S 


9 39, 


OVPETIFI O9TSIGT 
OOLE GP OG OvIGLed 
O07 90POELLE 


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“pre ; ‘soyouy 


06877 i 0F|60000079'00FLSL2|009260€ 0OFEOT! 


We 


“yar 4005 | ‘par 


00V9 


ay | 
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—- ————— 


-j|—= 


‘qoaog | -uleyg ‘pooy | ery | “oN 
“"AMOSVAN AUVADS P 
| j T 
8 V T 

I VOG GGG §9 T 

V i SOOT | OG 961 G T 

“ot Pe Lore: “S001 S88, tae) ae LT 
‘ea | ab. | *p& 4d “4b ‘yes | *pyy |3qa0 1d) +7, 


TION ‘$191 / prez || syufg}siren|snoyey spesy, sedig we 


-ren?) | 


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bide! 


‘AUOSVEN ALAOIT 


30 WEIGHTS AND MEASURES. 


DRY MEASURE. 


| Bushel. , Peck. - Quart. Pint. 
Lie BES uf qt. pt. 
7 2 64 
1 8 16 
1 2 


MOTION, OR CIRCLE MEASURE. 


| Oieele. Bina, Degree. Minute. Second. 
ee ee ee tg 21600 1296000. 

ont 30 1800 108000 

1 60 3600 

1 60 


DIVISION OF TIME, 


Year.|Months. Weeks. Days. Hours.| Minutes, | Seconds. 


Lae 52 | 365 | 8760 | 525600| 19536000 
1 4 
i 7 
1} 2 24e] 71446 86400 
1 60 3600 
1 60 


365 days 6 hours 


13 lunar months, 1 day, 6 hours } 1 year nearly: 


WEIGHTS AND MEASURES. 31 


TROY WEIGHT. 


~ Pound. | Ounce, | Pennyweight. Grain. 
lb. OZ. dwt. | or. 

1 12 240 5760 

1 | 20 | 480 

1 24 


By this weight gold, silver, jewels, and liquors, are weighed. 


AVOIRDUPOIS WEIGHT. 


Ton. anit Quarter.| Pound, |Ounce.|Drachm. 
se i ta A qr. lb. OZ. dr. 
| | 20 80 2240 | “1708 573440 
1 A 112 1792 28672 

1 | 28 448; 7468 

bi t6) (256 

1 16 


oe . . . . Se 
By this weight are weighed things of a coarse drossy nature, that are 
bought and sold by weight ; and all metals, butsilver and gold. 


APOTHECARIES WEIGHT. 


Pound. | Ounce. | Drachm. | Scruple. | Grain. 
1 | 2 96 288 5760 

| \ ae | 8 24 480 

1 3 60 

era 90 


~ By this weight apothecaries mix their medicines, but buy and sell. 
by avoirdupois weight. 


THINGS BOUGHT AND SOLD BY THE DOZEN, GROSS, &e. 


*Great | Comeinn Dozen. Particulars. 
gross. gross. d 
OZ. prs. 
g. gro. | Eo: a nn 
3 | 12 | 144 | 1728 
t 12 144 


WEIGHTS AND MEASURES. 


32 


ss 


+. 


“PeIMSLOUL OB SOULNL PUL S]JOA IO 19jVAs Jo sydop oY} Yor Aq Yoay XIs Jo OIMsvoML B st WOT] Y 
‘sia[vap esioy Aq posn A] uotULOO ‘seyout «Noy Jo eAnsvoUT ¥ SI pueYy VY 


€ I ” 
9g ro ¥ I 
SOT 9¢ . g T 
S81 96€ 0S ge II G 
P6S BUT GG FOL fi T ; 
9L°3 C6L OOT 99 GS ig T 
09286 O6L 000L 099 066 OF OT I ” 
O8006T 109869 0008 08S O9LT Were 03 8 » I 
—0b30049 ~—- |O8N06T 000FS OPSST O8G¢ 096 ~——s«{0nG 1G 52 € ait e 
© ; 
OOSPOPIL |009TO8E *l06008F 00sgTE 009G0L \00G6I j008F osp (f69 09 |06 Ir, 
HOO8SLEOTH IN WGLES9ET|OOOOON8TLT| OOOSFOFTI| OODITOSE.OO0GIGI LODOSSLT JOOSSATI STOSZIO09TS O0SLIO9 > Ir 
=] wa is 
> a a ee a, o, =e bie fone Le ORE on oie es 
& : P - Ea pe Oo Vege a ey 
8 | : a ee Sey. | 


“FUNSVAW DNOT 


COMPOUND APDITION. 33. 


COMPOUND ADDITION _ 


TreacuHeTn to collect numbers of different denominations 
into one total. 

Rule. Arrange the numbers so that those of the same 
denomination may stand directly under each ether, and 
draw a line under them. 

Add the numbers in the lowest denomination together, 
and find how many units of the next higher denomination 
are contained in their sum. 

Write down the remainder, and carry the units to the 
next higher denomination, and proceed thus to the end. 


EXAMPLES. 
FEDERAL MONEY. 
(1.) (2.) 
$ cts. m. $ cts. m. 
Li4- £4 3 39648 poh 4 5: QB 
186 67 5 376 --08- 6 
226 89 8 786 56 2 
164 40 9 310 Ss Rees. 
IRE NEY 967 50. 6 
Poe she 5) ee Ltt bay, ae 
——-—— 469-25. <8 
S57. 87 28 
Mee ee 2 
ie. og bon, 4k 


STERLING MONEY. 


(3.) (4,) 
Sic Benin ont oi iA gee Bi 
SE SS 68 475 18 6 
96°" 50-7 Ory 3 oak i LGA 9 
27 kB xd Oe 2 200 15 3 
330° 9@ B23 A968 °°: 7 
86.8 2 Toes AG2s ¥°10 
48 € 5296-38 Q94e <9°° Y 
—— Sao, |. 8 


4 


34 ‘ COMPOUND ADDITION. 


TROY WEIGHT. 


5.) (6.) 
lb. oz. dwt. gr. lb. oz. dwt. gr. 
4B ah 14 Ue 992.41 - 19ie83 
36.71To 48x23 107: 2:5 2589122 
84.140 17 20 2096392 kT eeBh 
68 4 13 13 200.8 16: 20 
Stab dhodd ob ve 3 B00u: oFatdeer 29 


96 enn): $64 lB tem 666 6 16.16 


cece ee 


7.) (8.) 
Ty -cwt. gts: (i og. sar. cut. gr. 1b. 
390 14 2-19 14 138 60,2 26 
680.19" 326 "13 9 45.3 18 
BPO ter dad Tos 1S 33 3 38 
pan 127-2 2°20 VFS 14 44 2 18 
BoG6. Th 3°13 VIS. 13 67 3 4 
376 14 2° 18 “15 13 58 1 16- 
APOTHECARIES WEIGHT. 
(9.) (10.) 
lb. oz. dr. sc. gr lb. 02. dr. 8. gr. 
Be Pt ees. Dp pe AL ae Cee 
SOU? Mar, « SMI tiie A 60°10" 66.75.18 
60. 1 tee 642 824 22 6 
AA SSS Ae BO. 1B ie Doda 
OO 38-6 Fa Te gore 6: 2.2.14 
CLOTH MEASURE. | 
(11.) (12.) 
yd. gr. na Ei. Fr. qr. na 
OOF ce oO 470" 2 3 
486 3 2 600 3 2 
386. 2 3 G00 -Y Se s1 
489 2 1 ree. aa 
643° 3°83 8 22 


500 
674 
387 
837 
444 


hhd 
5d 


¢€ 


COMPOUND ADDITION. 


WINE MEASURE. 


460° 28 


(13,) 4.) 
tun hhd. gal. gt. pt. hhd. gal. gt 
: RR ames PRE A 75. 40.3 
ie ere 44. 61. 2 
aS Hae aS Bho 4d op 
Si Sa eae! A Do, OU az 
Oo BO og pid 77, 40... 0 
ALE AND BEER MEASURE. 
(15.) (16.) 
gal, qt. pt. hhd. gal. gt. p 
55 3 I 80 47 3 2 
44. 2 0 63739. 2 0 
36.2. +8 SP 24 3. 1 
yds Mii | GHe29 25.1 
60... 3. SF PEELS 3 od 
44 2 1 87-36 1. 0 
DRY MEASURE. © 
(17.) (18.) 
bu. pk. qt. chal. bu. pk. gt. 
ys Sa CEOS OSPY 2.3 3S 
Cos #6 TESA RD LF 
See G G47: 9 26643 2G 
5° 2 6 434 28 0 4 
SEES 386 34° 1 3 
6 2 4 AST £2226 
LONG MEASURE. 
| (19.) 
deg. mu. fur. po. ft. in. bec 
SGO" 200" 7, "ov Zao Oe -2 
B02) SVE UR SE 961148 BD 
246 46 O24 Lae 9 oO 
Bee OL oe oo Vee 
3 gO 30 ioe ain 6 apse 


Sb 
. 


35 


36 COMPOUND ADDITION. 


LAND MEASURE. 


(20.) (21.) 
acr. TOO. pr. : acr. 00. pre 
645 3 29 866 * 3 "21 
74g = 2°98 643 2 20 
BOGS welt Bip. fap 
375-. 2-89 479 3 12 
269 1. 20 786 2 10 
317° 3.714 gag Bee gt se | 
TIME. 


years days hours min. sec. 
362.250 23 59 58 
4873 241 21. 40 47 
683; 264 20 56 $1 
a6%¢; 146.19 37 44 
486% 153 "16. 43 29 
(643 234-18: - 49. 48 


———s = 


_23. Find the amount of the following sums: £46 14s 
8d, £96 18s 6d, £47 18s 9d, £37 19s 10d, and £13 12s 
Ba uy, Ans. £243 4s 1d. 

24. In a contribution, A putin £7 14s 6d; B put in 
£1 8s 9d; C putin 12s 8d; D putin 6d 2qrs; E putin 
17s; and I’ 12s 4d: how much did they all pay ? . 

Ans. £11 5s 9d 2grs. 

25. If '7 men should each of them pay asum of £14 7s 
8d 2qrs, how much would they all pay ?. . 

Ans. £100 13s 11d 2qrs. 

26. A man had three sons: John was 6 years 4 months 
_ old; George was 8 years 6 months and 12 days old; 

James was 18 years old. What was the age of all of 
them ? Ans. 32 yrs. 10mo. 12 days. 

27. Bought a quantity of goods at New York to the 
amount of £384 17s 8; paid for carting to the dock, 12s 
8d; paid for freighting the same to Albany, £2 17s 8d; 
‘then paid for carting the same to Geneva, £7 Is 10; and 
my own expenses were £6 14s 9; how much do the 
goods stand mein at Geneva? Ans. £402 4s 7d. 


COMPOUND SUBTRACTION, | 


28. If A should pay 15s 8d; B pay twice as much; C 
pay twice as much as B; D pay 7s 6d; and E pay as 
much as all the others; how much money “would they all 

ay? : Ans. £11 14s 4d. 

29. If I have a silver tankard that weighs 3 pounds 7 
ounces 16 pennyweights and 4 grains, and a dozen silver 
table-spoons, weighing | pound and 12° grains, a sugar 
bowl that weighs 9 ounces and 10 pennyweights, and six 
‘tea-spoons Veighing 17 pennyweights each ; how heavy 
do they all weigh ? Ans. 5 lbs. 10 oz, 8 dwt. 16 girs. 

30. Suppose T have five barrels of potash, whose weight 
is as follows: the first'weighs 3 cwt. 1 qr. and 12 lbs.; the 
second, 2 cwt. 3 qrs. and 26 lbs.; the third, 3 cwt. 3 qrs. 
and 6 thar: the fourth, 3 cwt. and 17 lbs.;°the fifth, not 
being well ‘packed, would not weigh more than 2°cwt. 24 
Ibs.: what is the weight of the whole ? 

Ans, 15 cwt. 2 qrs. 1 1b. 


COMPOUND SUBTRACTION. 


TeacnrtnH to find the inequality between numbers of 
divers denominations. 

Rule. Having arranged the numbers so that the smaller 
may stand under the greater, subtract each number in the 
lower hne from that which stands above it, and write - 
down the remainders. When any of the lower denomina- 
tions are greater than the upper, increase the upper num- 
ber by as many as make one of the next higher denomina- 
tion, from which take the figure in the lower line, and set 
down the remainder, carry one to the next number in the 
lower line, and subtract as before. 


EXAMPLES. 
FEDERAL MONEY. 


(1.) (2.) 


& cts. m. $ cls. mM. 
G57.) 46.6 99. 48° +39 


367 37. 4 86,...00) 37 


eee ee 


38 


COMPOUND SUBTRACTION. 
STERLING MONEY. 
(3.) (4, 
ty 8 Oe Ls. d. gr. 
100 1p 10723 4 6. 4 2 
GO ta Oo a CM: st og 
TROY WEIGHT. 
(5. (6) 
lb. oz. dwt. gr, 5 a td. 2-2, Abs a. 
AD ce hOr so 2 5G eet Oy. Be Tobie 
= Sos SMe Ge ie OOo, ph ee ¥ 
AVOIRDUPOIS WEIGHT. 
(7) (8.) 
ton cwt. gr. 1b. oz. dr. cwt. gr. 1b. 
rT tee Sa Sco eed lame QB icBineB 
62° 16 21426 15 7 54.2. .26 
APOTHECARIES WEIGHT. 
(9.) 7 (10.) 
lb. “oz. dro scx? g7. lb. oz. dr. sc. gr. 
GT 1 96" r2. 1G a”. Gy Oo po 
os 1 Sede Mahl phi: “mane, ao 0 & & 
CLOTH MEASURE. 

(11.) (12.) (13.) 
yd. gr. na. EF. qr. na. EE. qr. na. 
65 3 3 189s 2-23 60 3 3 
25 2° 3 ROD ie £3. 2 

WINE MEASURE. 
(14.) (15.) 


tun. hhd. gal. qt. pt. 
32 OSes MO oI Ss CA I 
G46 2). 60 TR ee 


hhd. gal. gt. pt. 
R thacg ee is ak 
69°" 242 .0 


COMPOUND SUBTRACTION. 


ALE AND BEER MEASURE. 


Pty (17.) 
hhd. gal. qt. pt. hhd. gal. qt. pt. 
89° 46 2 1 675° 60 2. I 
Of ooo. 536 50 3 0O 
DRY MEASURE. 
(18.) (19.) 
gr. bu. gal. gt. chal.. bu. gal. gt. 

38 4 3.3 ey aoe ay Bee 
365 5 2 SOT 2 34 5. 12 
LONG MEASURE. 

(20.)° (24:) 
deg. m. fur. p. Ji in, b.C. WK Jura De oft 
BEE 3630 6.24 9 AO) 8 Be EB yah 
643 60 4. 8 i ii. 2 Oh ES a hiigh 20 

LAND MEASURE. 3 
(22.) (23.) 
acr. T00. per. acr. 700. per. 
§'7575. 25011 C9 ead ae: 
484 3 165 69° 3 8 
TIME. 
(24.) : (25.) 
“rs: Ga. "hf. “Mm. Sec. YS. Gs BPs, Ube. SEC. 
437 116 18 44 36 G7. 3007 21 2O0- 0 
Shh e100." Or Dae to 18 364 23 46 56 
SOLID, OR CUBIC MEASURE. 
(26.) (27.) 
cord fl. Mm ton fl. Ms 
45 118 136 24 3%. 184 
GO 2S7< 9G 37 24 1712 


39 


40 COMPOUND MULTIPLICATION. 


28. Borrowed £50 10s: paid again at ofe time £17 _ 
11s 6d; and at another time £9 4s 8d; at another time 
£17 9s 6d; and at another time 19s Gd 2 qrs. How much 
remains unpaid Q Ans. £15 48 9d 2 qrs. 

29. Borrowed £100, and paid in part as follows, viz., at 
one time £21 11s 6d; at another time £19 17s 4d 2 qrs; 
at another time 10 dollars, at 6 shillings each; and at 
another time 2 English guineas, at 28 shillings each, and 
- 2 pistareens, at 14d 2 qrs each; how much remains due, 
or unpaid ? Ans. £52 128 8d 2 grs. 

30. A, B, and C, drew their prize- money as follows, 
viz., A had £75 15s 4d: B had three times as much as A 
lacking 15s 6d; and C had as much as A and B both ; 
how mifth fea ee Ans, £302 5s 10d. 

31. I lent John Paywell 1000 dollars, and afterward 
lent him 26 dollars and 45 cents more. He has paid me 
at one time 361 dollars 40 cents, and at another time 416 
dollars 9 cents, besides a note which he gave me on Peter 
Trusty for 143 dollars 90 cents; how stands the balance 
between us ? And. $105,06 cts. my due. 

a Paid A Bin full for E F’s bill on me for £105 10s, 
viz.; I gave him Paul Jones’ note for €15 14s 9d; John 
Cook’ s note for £30 0s 6d; an order on Sam Patch for 
£39 11s; the rest 1 make up in cash. I wish to know 
what sum will make up the deficiency ? — 

Ans. £20 3s 9d. 


COMPOUND MULTIPLICATION 


Is the multiplying of numbers of different denominations 
by a simple figure or figures, whose product shall be 
equal to a pr oposed number or numbers, 

Rule. Write the multiplier under the lowest denomina- 
tion of the multiplicand; multiply every number of the 
multiplicand by the multipher, and bring the several pro- 
ducts as they occur to the next higher denomination ; 
write down the remainders, and carry the integers to the 
next product, 


COMPOUNIP MULTIPLICATION. 


EXAMPLES. 
STERLING MONEY. 


TROY WEIGHT. 

(5.) 16) 5p 
lb. oz. dwt. gr. lb, oz. dwt, gr. 
76 10.14 23 4°62 8.4 

9 see! 8b 


feeeaneee ae 


AVOIRDUPOIS WEIGHT. 
(7.) (8.) 
ton. cwt. gr. lb. oz. dy. ID. Oz aN 
Pit Vis tio 108712 96 oh “be 
5) 6 


te re er eee 


APOTHECARIES WEIGHT. 
(9.) (10.) 
Coot Gee APC 8e, (BT's * 1S 2. Bi. Sle... P, 
44 4 4 2 16 16:°° 8 3S bs 14 
fone 12 


re a Em et eee near 


ee eee pee en Se 


| CLOTH MEASURE. 

(11.) (12.) (Fax) °< 
yd. gr. na. Ei. E. qr. na. ELF r. qr. na. 
Si eed 64 2 2 iG leet 

4 6 “8 


ee ee ee en ee re 


42 


14.) 


tun hhd. gal. qt. 


3764 1 60 8 


a 


ce ne ee 


* 


COMPOUND MULTIPLICATION, 


WINE MEASURE. 


pet. 
1 
11 


(15.) 
hhd. gal. qt. pt. 
900 45 1 1 


ww: 


ALE AND BEER MEASURE. 


(16.) 


hhd. eal gt. pt. 


(17.) 
hhd. gal,gt. pt. 


Py 8 ra Dat.” 40° "a we 
16 14 
DRY MEASURE. 
(18.) (19.) 
gt. bu. gal. qt. chal, bu. gal. qt. 
A rae aay Pie COs wl Seek poke 
18 6 
LONG MEASURE. 
(20.) (21.) 
deg. Mh, SU, Dio fbs ots D220 m. fur, p. ft. 
Bae a ae eG es PL) ee ab? Pee: Spree Fs ales be 
19 20 
LAND MEASURE. 
(22.) (23.) (24.) 
acr. 700, per. acr. 700. per. acr. TOO. per. 
1000 3 14 10 SR 55 2 8 
14 15 Rin -8 16 
SOLID, OR CUBIC MEASURE. 
(25.) (26.) 
cord * ft. in. ton ft. in, 
240 2 en by i: "ee 666 83 1726 
70 60 


eee ee ee 


COMPOUND MULTIPLICATION. 43 


TIME. 
(27.) eS HOR) 
yo. da. hr. m. sec. OT.” Ud, TTY Ty SOG: 
Oo 218 41 40: °6! QE SG e oO Lee 
35 86 


: 4 
a SS « a SR ee 


oe ee pen | 


Notr. When the multiplier is acomposit number, and greater than 12, 
take any two such numbers as, when multiplied together, will exactly pro- 
duce the given quantity, and multiply first by one of those figures; and that 
product by the other, and the last product will be the answer. When no 
two numbers, multiplied together, will exactly make the multiplier, you 
may multiply by any two whose product will come the nearest ; then mul- 
tiply the upper line by what remained ; which, added to the last product, 
gives the answer. 


BILL OF PARCELS. 


Boston, June 15th,, 1839. 
Mr. Peter Dow, ~ 
Bought of Geo. Smith 5 Co. — 


8 pairs worsted hose, at 4s 6d, $6 00 
5. do. thread do. tae 2d, 2 64 
3 yds. kerseymere, 14 7 00 
6 do. muslin, RN TD ae is cos 4 16 
2 do. tammy, ti Ts Sd, 0 56 
4 shawls,  - ve “fe Gd, 5 00 
644 yds. nankins 28 21 50 
32 ells mode, ft SS -16 00 
28% yds. calico, oo Os Ad; 11 08 
2 gross gilt coat buttons, “ 18s 6d, 6 17 
3 pieces russel, “¢ 34s eee OO - 
2 do. muslin, #61308, 4 10 00 
25 yds. Irish linen, a e835 
284 do. stormount calico, : fics: Gd, 11 88 
284 do. red do. fi .23.2d, 10 29 
1 piece durant, : 56s, 9 33 
2 pieces blue shalloon, © “ 57s 6d, 19°17 
50% yds. dimity, 26. Ga, 21 04 
3 pieces persian, i Bag, 42 00 
Amount at 6s to the dollar, $229 15 
8s - - 171 86 
7s 6d - 183 32 


44 ‘ COMPOUND DIVISION. 


29. What is the weight of 7 hhds. of sugar, each weigh- 


ing 9 cwt. 3 qrs. 12 lbs 4 Ans. 69 cwt. 
30. What is the weight of 6 chests of tea, each weigh- 
ing 3. cwt. 2 qrs. 9 lbs? Ans. 21 cwt. 1 gr. 26:/bs. 
31. How much brandy in 9 casks, each containing 41 
gals. 3 qts. 1 pt.? Ans. 376 gals. 3 gts. 1 pt. 
32. In 35 pieces of cloth, each measuring 27 yds. 3qrs., 
how many yards ? Ans. 971 yds. 1 qr. 
33. In 9 fields, each containing 14 acres 1 rood and 25 
prs how many acres ? Ans. 129 acrs. 2 roo. 25 pr. 
34. In 6 parcels of wood, each containing 5 cords and 
96 feet, how many cords ? Ans. 34 cords 64 feet. 


35. A gentleman having 18 silver spoons, each weigh- 
ing 2 oz. 15 dwt. 11 ers.; also 24 tea-spoons, each weigh- ° 
ing 10 dwt. 14 ers.; and 2 silver tankards, each weighing 
21 oz. 15 dwt. Pray, what is the weight of the whole? 

Ans. 8 lbs. 10 oz. 2 dwt, 6 grs. 


tae 


COMPOUND DIVISION 


eo 
TEACHETH to find how often one number is contained in 
another of different denominations. 


Rule. Begin at the left hand, and divide each denomi- 
nation by the divisor, setting down the quotients under 
their respective dividends. But if there be a remainder 
after dividing any of the denominations except the least, 
find how many of the next lower denomination it is equal 
to, and add it to the number, if any, which was in. this 
denomination before, then divide the sum as usual, and so 
on, till the whole is finished. 


The method of proofis the same as in Simple Division. 


STERLING MONEY. 


(1.) (2.) 


ee i Bags HR L£ $d, gr. 
2)64 18 6 3)3TSe 8 139TH S 
on > Deeg 2 ae gee 


ee rr ee on ee em ee ee 


COMPOUND DIVISION. 45 


TROY WEIGHT. 


(3.) (4.) 
lb. oz. dwt. gr. lo: 62, dwt. gt. 
8)44 8 12 4 7)75 3.16 19 
AVOIRDUPOIS WEIGHT, 
(5.) (6.) 
ton cwt. gr. lb. oz. dr, 1b. oz. dr 
9)48 16 1 14 13 12 11)14-3 3 
APOTHECARIES WEIGHT. 
(7.) (BO) ts 
1b. Ute AT. 3e.-27. lo: Some ar. ac. Br. 
12837 25 Gat. Th NS aged RON gas ame 
CLOTH MEASURE. 
(9.) (10.) egaae 
yd. qr. na EE. gr. na. EFI. qr. na 
6)74. 2 8 7)88 1 1 S)77 2 1 
WINE MEASURE. 
(12-5 (13.) 
tun hhd, gal. qt. pt. hhd. gal. qt. 
1099 2 66 3 1 9777 44 1 
ALE AND BEER MEASURE. 
Pray: (15.) 
Phhd. gal. qt. pt. hhd, gal. qt. pt. 
11)83 63 1 0 S:cl 


6)911 45 


Re 


ST ern sey SR wae 


RE ESE OED, 


46 


COMPOUND DIVISION. 


DRY MEASURE. 


(16.) (17,) 
bu. gal. qt chal. bu. gal. qt. 
734-63 9)643° 33° 4° 3 
LONG MEASURE. 
~ (18.) 19.) 
eg nmi fur. p." ft. in... m. fur. %. 
3)47. 49° 6 27.°8 10 2° 7)37° 4°30 
LAND MEASURE. 
(20.) (21.) 


acr.. T00. per. 
12)974... Lig B7 


a eS 


— 


SOLID, OR CUBIC MEASURE. 


(22.) 


ton ft. tm. 
11)91 39 144 


ened 


- (24,) 


8s niin Albin, 


(23.) 
ton fe. an. 
11)684 “17 1727 


TIME. 
(25.) 


mM. Se, yr. da. hr. m. 


12)365 113. 11 39 49 CNY aoe Deak Mods 


ES SL 


(a ee ane 


Lt SN pee 


COMPOUND DIVISION. 44 


Notr. Wheft the divisor is large, and not-a composit 
number, you may divide by the whole divisor at once, after 
the manner of long division, thus: 


(26.) 
= Ae Ape 
37/46 1 T1(£1 
37 
9 
20 
37)181(4s : 
148 ms 
33 
12 
37)407(11d Ans. £1 4s 11d. 
407 ' 


(27,) 
lb. oz. dwt. 
24)26 1 d(1Ib 
24 
2 
12 
24)25(loz. 
24 
iL 
20 


24)25(1dut. 
24 


est 


1 
24. 


24)24(1gr, Ans. 11d, loz, ldwt. 1gr. 


“ 


48 DECIMAL FRACTIONS. 


28. Divide 4 gallons and 2 quarts of brandy equally 
among 144 soldiers. ‘Ans. 1 gill a piece, 
2). “Bought 12 silver spoons, which together weighed 
3 lb. 2 oz. 13 dwt. 12 gr., how much silver did each spoon 
contain ? Ans. 3 oz. 4 dwt. 11 grs. 
30. Bought 17 ewt. 3:qrs. 19 Ibs. of sugar, and sold out 
one-third of it; how much remains unsold ? 
Ans. 11 cwt. 3 gr. 22 lbs. 
31. From a piece of cloth containing 64 yards 2 nails, 
a tailor was ordered to make 9 soldiers’ coats, which took 
one-third of the whole piece; how many yards did each 


coat contain ? Ans. 2 yds. 1 gr. 2 na. 
32. If a man spends £74 14s 6d a year, what is that per 

calendar month ¢ Ans. £5 19s. 64d. 
33. The Prince of Wales’ salary is €150,000 a year; 

what is that per day? Ans. £410 19s 2d. 


34. A privateer takes a prize worth 12,465 dollars, of 
which the owner takes one-half, the officers one-fourth, and 
the remainder is equally divided among the sailors, who 
are 125in number; how much is each sailor’s part ? 

Ans. $24,93 cts. 


DECIMAL FRACTIONS. 


A Dectmat Fraction is that whose denominator is an 
unit, with as many ciphers annexed to it as the numerator 
has places, and is usually expressed by writing the nume- 
rator only, with a point before it, called the separatrix ; 
thus, * rvs y's are decimal fractions, and are expressed 
by ,5 ,25 ,125 respectively. The figures to the left hand 
of the separatrix are whole numbers : thus, 4,5 yards is 4 
yards and 5 tenths, or one half of another yard. 

Ciphers placed to the right hand of decimals erative’ no 
alteration in their value; ,5 ,50 ,500, &c., are decimals of 
the same value, being each equal to 3; but when placed 
to the left hand, the value of the fraction is decreased in a 
tenfold proportion; thus, 5, 05 ,605, &c., are 5 tenth parts, 
5 hundredth parts, 5 thowsandel parts, respectively. 


The different value of figures will appear plainer by the 
following 


DECIMAL FRACTIONS. 49 


TABLE. 


Integers. Decimals. 
a a aid 


2; 
2 40, 3.2 
2 O02 
20 84020 0 2 
20° 0°0 8200 0 2 
2 0 0 030030 @ 0 0.2 
2000000,600002 
2000000 0,00000 0 2 
200009000,00000002 
Here ete geste ae om 
pePSsexp Pop rast soe sp Seg 
pe eS ea ee Pr os eet Ga 
ft OF t's 3D ye mim Se 3 2.7 
a oe ane ete on ee 
Bee Be BEe SSS 
ofr 9 6 ask S Eas 
ane sia, @ Pee Fs 
BS. Fo a 
aah caahuael ad oo 8 oe 
o a ey Ss 
wR 


From the above table it appears that as whole numbers 
increase in a tenfold proportion from units to the left hand, 
so decimals decrease in ‘the same proportion to the right : 
and that in decimals, asin whole numbers, the place of a_ 
figure determines its relative value. 


Example for writing Decimais. 


Five tenths, - - - - - - re) 
Five hundredths, bel - - 05 
Five thousandth, - - - - . ,005 
Five hundred thousandths, —- - - ,0005 
Fifty-three thousandths, = ee oak eS 


Five and fifteen hundredths, - - « 5,15 


ADDITION OF DECIMALS. 
“Rule. Place the given humbers so that the decimal 
points. may. stand ‘directly under each other; aie a ag 
§ 


50 ADDITION AND SUBTRACTION..OF DECIMALS. 


in whole numbers, and point off so many places for deci- 
mais to the right as are equal to the greatest number of 
the decimal places i in any of the given numbers. 


EXAMPLES. 

(1.) (2.) (3.) 
263,51 42,23 2,1 
149,28 18,47 5 
293,53 9,3 26,17 
184,59 52,384 a 
129,4 2,1 5, 

1020,31 124,484 | 34,47 


4, Required, the sum of twenty-nine and three-tenths, 
three hundred and seventy-four and nine millionths, 
ninety-seven and two hundred and fifty-three thousandths, 
three hundred and fifteen and four hundredths, twenty- 
seven, one hundred and four-tenths. Ans. 942,993009. 

5. Required, tue sum of ten dollars and twenty- -nine 
cents, ninety-three cents and three mills, nine cents and 
six mills, and two dollars and eight mills. 

Ans. $13,32 ets. '7 m. 


SUBTRACTION OF DECIMALS. . 
. Rule. Place the given numbers so that the decimal 


points may stand directly under each other, ard poitstl off | 
the decimal places, as in Addition. 


EXAMPLES. 
(1.) (2.) (3.) 
From 219,42 87,26 ene ae ee 
Take 184,38 19,4 ee 9,375 
35 04 67,86 -, 47,626 


—_—_— 


oe cee 


4. From two thousand and sixteen hundredths, take 
one thousand and four and four millionths.””~ 


° Ans. 996,59996, 


5. From seaney® -four Dieusend nine hundred and. nine 
and one-tenth, take fourteen thousand and twenty-nine 


thousandths, 


Ans. 10909,071. 


t, 


MULTIPLICATION OF DECIMALS. 5i 


6. Take eighty-five and seven hundred and thirty-seven 
thousandths from one hundred. oO Ans. 14,263. 

7. From five hundred and thirty-one dollars two cents, 
take one hundred and seventeen dollars three cents and 


four mills, Ans. $413,98 cts. 6m. 
8. From ten dollars and eight cents, take one dollar and 
three mills. _ Ans. $9,07 cts. 1m. 


MULTIPLICATION OF DECIMALS. 
Muutipiy exactly as in whole numbers,.and_ from, the 
product cut off as many figures for decimals to the right 
hand as there are decimals in both factors; butif the pro- 


duct should not have so many, supply the defect by pre- 
fixing ciphers. 


EXAMPLES. 

(1.) (2.) 
Multiply 36,5 29,831 
by 7,27 — ,952 
2555 59662 

730 149155 

2555 268479 
265,355 : ~-28,399112 

(3.) (4.) (5.) 

Multiply ,285 285 529 

by” ,8 ,003 tr 
,2280 000855 ‘s0e0 


a eee: —_— ——— 


Note. To multiply decimal fieeiie by 10, 100, 1000, 
&ci, is only to remove the separatrix ‘so many i 
towards the right as there are ciphers. 


(10, . )73,62937. 
106, 7 (786,293 
Thus: 7,362937 2 1000, 71362,937% ° 
L.10000, 573629;37. 
6. “Maltiply ¢ two chowadhid and four and two-tenths fn 
twenty-seven. Ans, 54113,4, 


CT PCIiTV Ac ILL TRIE 


YERSI] } L} al KS 


52 DIVISION OF DECIMALS. 


DIVISION OF DECIMALS, 


' Rule. Divide as in whole numbers, and from the right 
hand of the quotient point off as many places for decimals 
as the decimal places m the dividend exceed those of the 
divisor. Ifthe places ofthe quotient are not so many as 
the rule requires, supply the defect by prefixing ciphers. 
If at any time there be a remainder, or the decimal-places 
in the divisor are more than those in the dividend, ciphers 
may be annexed to the dividend, and the quotient carried 
to any degree of exactness. 


= - EXAMPLES. 
(Eiye ets yl x bra 30(22) 
92),863972(,009391 ,853)89,000(104,337 
828 853 
359 3700 
276 3412 
837 2880 
828 2559 
92 3210 
92 2559 
6510 
5971 
539. 
3. Tivide 803 by, 22 Ans. 3,65 
4. Divide 8,03 by 2,2° ; 365 
5. Divide ,803 by 22 ,0365 
6. Divide 80,3 by ,22 36,5 
te Divide 80,3 by 2,2 io Bi6ho5 
8. Divide 222 by 865.» 60821. 


To reduce quantities of several denominations to a decimal. 


Rule. Place the several denominations above each other, 
letting the highest denomination stand at the bottom; then 


o> 


owe ay 4 


~. vied 


REDUCTION. 53 


divide each denomination (beginning at the top) by its 
value in the next denomination, the last quotient will give 
the decimal required. 


EXAMPLES. 
1. Reduce 12s 6d 3qrs to the decimal of a pound. 
4] 3, 
121 6,75 
20 | 12,5625 
| eee ee aeons 
| | ,628125 
2. Reduce 15s 9d 3qrs to the decimal of 4 pound. 
Ans. 5790625. 
3. Reduce 9d 3qrs to the decimal of a shilling. 
Ans.,,8125. 


Note. When the shillings are even, half the ear. 
with a point prefixed, is their decimal expression; but if 
the number be odd, annex a cipher to the shilling, and 
then, by halving them, you will have their er ex- 
pression. 


REDUCTION 


. Tracners to change numbers from one denomination 
to another, without losing their value. 

When numbers of a higher denomination are to be re- 
duced to a lower, it is called Reduction Descending, and 
it is performed by Multiplication. When numbers of a 
lower denomination are-to be brought to a higher denomi- 
nation, it is called Reduction Ascending, and it is perform- 
ed by Division. 


REDUCTION DESCENDING. 


Rule. Multiply the highest denomination by as many of 
the next less as make one of the greater, adding to the 
product the parts of the same name, and so on to the last. 


54. , REDUCTION. 


"EXAMPLES, 


ABS £987 14s 6d 3 qrs., how many farthings ? 
987 14 6:3 
20 


19754 shillings. 
12 
237054 pence. 
4 


948219 farthings. 


2. In 11 oz. 13 dwt. 13 grs., how many grains ? 
Ans. 5605 grs. 
3. In 13 cwt. 3 qrs. 21 Ibs., how many pounds ? 
Ans. 1561 lbs. 
4, In 57. years, how many hours, allowing each year to 
be 365 days 6 hours ? Ans. 499662 hours. 


REDUCTION ASCENDING. 


Rule. Divide the given number by as many of that de- 
nomination as make one of the next higher, and so on to 
the denomination required, and the last quotient, with the 
several remainders, if any, will be the answer. 


EXAMPLES. 


1. In:46788 farthings, how many pence, shillings, and 
pounds? 


4)46788 
12)11697 

2,0)97/,4---9 

48--14--9 


Ans, £48 14s 9d. 
1. In 900 far things, how many pounds ? 
Ans. £0 18s 9d. 
3. In 243648 farthings, how many dollars, at 6 shillings 
each % : Ans. 846 dollars. 
4, Reduce 13776 pence to guineas, at 28s per guinea. 
Ans. 41 guineas. 


REDUCTION. 55 


5, In 62304 farthings, how many pistoles, at 22s each 4 
Ans. 59 pistoles. 

6. In 24396 pence, how many shillings, pounds, and pis- 
toles? Ans. 2033s, £101 133, and 92 pistules and 9s over. 


Questions promiscuously placed: 


1. Suppose A and B were to travel from Vergennes, in 
the State of Vermont, to Geneva, in the State of New 
York, the* distance being 300 miles; A steps 2 feet 6 
inches each step, and B but 2 feet 4 inches; how many 
more steps must B take to perform his journey than A ? 

Ans. 45257. 

2. It is supposed the wars of Bonaparte, in 20 years, 
caused the death of 2,000,000 of persons; how many was 
this per hour, allowing the year to contain 365 days 6 
hours! © Ansa] REAR OS 

3. A goldsmith having 15 ingets of silver, each weigh- 
ing 2 Ibs.'7 oz. 3 avers: whidh Hheiowishad tomnake Hid 
bowls of 2 Vhs! 8°Hz., faikueds of 1 lb. 10 oz. salts of 11 
oz., and spoons of 1 oz. 15 dwt , and of each an ear 
number; how many will there belbfréach sort? Ans. 7 

A, If sound, uninterrupted, moves 1142 feet in one 
second, how long would it be in passing from the sun to 
the earth, the distance being estimated at ‘95,000 nes of 
miles?) © Ans. 13 yrs. 338 d. 16h. 10 m. 22 see. Tas 

5. Admit a ship’s cargo from Iondon to be 250 pipes, 
130 hhds., and 150 half Hhae , how many gallons in all, al- 
lowing every pint to be a.pound, and what is the ship’s 
‘burden? Ans, 44415 gals., and 158 tons, 12 cwt. 2 qrs. 


RULES OF PROPORTION. 

Havine introduced the fundamental principles of Arith- 
metic, we come now to the rules of proportion. Under 
this head may be classed the following, viz., Multiplica- 
tion and division of fractions, reduction of fractions, 
reduction of currencies, interest, banking, commission 
insurance, ratio, simple and compound proportion, simple 
and compound proportion in fractions, conjoined propor- 
“tion, discount, profit and loss, barter, partnership, com- 
mercial exchange, tare and tret, equation of payments, 
mensuration or practical geometry, &c, &c., and in fact 


4 


56 RULES .OF PROPORTION. 


almost every thing where.multiplication and division are 
concerned. It is very important, not only to know how to 
solve propositions under the various.rules, but also how 
they may be solved most expeditiously. The principle of 
cancelling is doubtless the greatest desideratum for facili- 
tating arithmetical problems that has ever been introduced 
into the science of mathematics. The object of which is, 
to acquaint the scholar with a principle by which peculiar 
expedition is attained in the solution of such sums as 
involve in their operation both multiplication and division. 
This principle is founded on the following facts: First, 
The value of any quotient depends on the ratio, or relative 
size of the numbers divided; that is, if the dividend be 
five times as large as the divisor, the value of the quotient 
is five, and if it be eight times as large, the value is 8, &e. 
Second, If two’ or more numbers are to be multiplied 
together, and their product divided by any other number, 
the true result is obtained by first dividing one of these 
numbers, by the dividing number, and then multiplying 
the quotient by the remaining number or numbers, . Thus, 
if it be required to multiply 8 by 4 and to divide the pro- 
ange by 2, first divide 8 by 2.and multiply the quotient by 
4, thus, 3--2= 4, and 4+4=—16. The advantage-of this 
process will be more obvious if we take large numbers. 
Suppose we wish to multiply 1728 by 16, and to divide 
the product by 144, the usual process would be thus : 


1728 By first dividing 
16 the operation is 
—_—— much abbreviated, | 
10368 thus: 
1728 144)1728(12 
were 1728(16 
144)27648(192 —— + 
144 0000 192 
1324 
1296 
288 . : 
288 


000 


RULES. OF PROPORTION. 57 


By the usual method, 46 figures are required, by the 
other. only 22: There is, still another advantage. The 
scholar can see ata glance that 144 is contained in 1728 
twelve times, and that 12 times 16 1s 192;-s0 thatan ope- 
ration, which is long and protracted, is often reduced 
nearly or quite toa mental operation. Third, When any 
large number is to be divided by the product of two or ~ 
more smaller numbers, it may be divided by each number 
separately. This needs no explanation, it is the same as 
dividing by the component parts of any number, instead 
of the number itself. Fourth, When the: operation is of 
such a nature as to_require the product of several num: 
bers to be divided by the product of several other num- 
bers, these numbers may be divided before multiplication, 
and their quotients used, instead of the numbers them- 
selves. For illustration, suppose the product of 36 and 
42 is to be divided by the product of 6 and 7, the usual 
mode of operation would be as follows, viz : : 


AQ 6 :. 
36 a 
252 42 Divisor. 
126 


1512 Dividend. 
42)1512(36 The required 
126 quotient. » 
202 
202 


eee 


000 


But by the preceding fourth principle, 36—.6=6, and 
42-_7—6, and 6+6—=36. Ans. In this example the divi- 
sors are, as it were, expunged or lost, since they divide 
without remainder. 

But for further illustration, suppose it be required to 
multiply the numbers 72, 40, 84 and 36 together, and to 
divide the product successively by 12, 8,144 and 7. Itis 
desirable to arrange these numbers so that they may be 
conveniently compared with each other. We will adopt 

5 


58 RULES OF PROPORTION. 


the following mode:°We will place the numbers whose 
product is to form a dividend, on the right hand side of a 
perpendicular line, and those whose pr ‘oduct is to form a 
divisor, on the left hand side, thus : 


144 
40 
7 | 84 
12 | 36 


Now, by the fourth and last principle laid down, T can 
divide 144 in the divisor and 72 in the dividend by 72, 
without a remainder, and obtain 2 in the divisor and 1 ti 
the dividend; thus : 


I can alsp divide 40 in the dividend and 5 in the divisor 
by 5, and obtain 8 in the dividend and 1 in the divisor, 
thus: 


2|1 
1/8 
7 | 84 
12 | 36 


Again, I can divide 84 in the dividend and 7 in the di- 
visor by 7, and’ obtain 12 in the dividend and 1 in the 
divisor, thus : 


Again, I-can divide 36 in the dividend and 12 in the 
divisor by 12, and obtain 3 in the dividend and 1 in the 
divisor, thus : 


| 
! 
Again, I can divide 8 or 12 in the dividend and 2 in the 


GENERAL RULES. 59 


divisor by 2,.and obtain 4 or Gin the dividend and 1 in 
the divisor, thus : 


{. 
4 
12 
3 

It is now evident that the division can be carried no far- 
ther without remainder. The next step, therefore, is to 
divide the product of the numbers remaining on the right 
hand side of the line, by the product of unse on the left. 
The product of those on the right is 4-4+-12+3—=144 ,and of 
those ou the ieft bee eens therefore 144: 1—144, 
the number required. The same result would have been 
obtained by multiplying the numbers together on the right 
hand side of the line, and dividing their product by the 
product of those on the left hand side, previous to can- 
celling. In the above example, as the numbers have been 
cancelled they have been omitted, and a new statement 
made. This is by no means necessary. One statement 
is sufficient. It will be noticed that in every instance 
division is effected without a remainder. Such must al- 
ways be the case. 

The following rules will be found a competent guide 
for the learner in all operations of cancelling: 


1 
1 
1 
1 


GE Meee RULES. 


Rule ist. Draw a perpendicular line; observe this line 
represents the sign of equality. On the richt hand side 
of this line, place dividends only; on the left hand side 
place divisors only. Having placed dividends on the right 
and divisors on the left, as above directed, 

2d. Notice whether there are ciphers both on the right 
and left of the line; if so, erase an equal number from 
each side. 

» 3d° Notice whether the same number stands both on 
the right and left of the line; if so, erase them both. 

ath. Notice, again, if any ‘number on éither side’of ‘the 
line will divide any number on the opposite side, withcut 
a remainder; if so, divide and erase the two numbers, 


60 VULGAR FRACTIONS. 


retaining the quotient figure only on the side of the larger 
number. 

5th. See if any two numbers, one on 1 éach side, can be 
divided by any assumed number without a remainder; if so, 
divide them by that number, and retain only their quo- 
tients. Proceed in the same manner, as far as practicable, 
then, 

6th. Multiply all the numbers remaining on the right 
hand side of the line for a dividend, and those remaining 
on the left for a divisor. 

7th. Divide, and the quotient is the answer. 

Note. If only oné number remain, on either side of the 
line, that number is the dividend or raids according as 
it stands on the right or left of the line. The figure “Lig 
not regarded in the operation, because it avails nothing 
_ either to Racaae a or divide Py: 


VULGAR FRACTIONS. 


Fractions or broken numbers, are expressions for any 
assignable part of an unit, and. are represented by» two 
numbers, placed one above the other, with a horizontal 
line drawn. between them. The number above the hori- 
zontal line is called the numerator, and that below the line 
the denominator. The denominator shows how many parts 
the integer is divided into, and the numerator shows how 
many of those parts are meant by the fraction. 

Fractions are either proper, a compound, or 
mixed. 

ist. A proper fraction isvhen the numerator is less 
than the denominator, as 3, $5 Fs 49) Tres losTsseer 

2, An improper fraction is when the pubes ator is either 
equal to or greater than the denominator, as th Fs eee 

3. A compound fraction is a foackiom of fracas and 
lola Py the word of, as 3 of 2 of £ of =°- of $ of 4? of 
p Of yoo 

4, A mixed number or fraction is composed of a whole 
nut bee and a fraction, as 83, 127, 17282, 9999999.9., 
Mixed numbers may be reduced to improper fractions, by 
multiplying the whole number by the denominator of. the 
fraction, and to the product. add the numerator for a new 
numerator, and place it over the denominator. 


ADDITION AND SUBTRACTION OF FRACTIONS. 61 


ADDITION AND SUBTRACTION OF FRACTIONS. 


Addition and. subtraction of fractions: are easily per- 
formed, if the fractions have common denominators. . Ex- 
amples: +,2,3,.4, added together, equal 12; here we 
merely. add the, numerators, and place their common de- 
nominator under the sure of the numerators for the an- 
swer. Again, add 4, 4, $, 354, together=2 —27; adding 
the numerators, we have 25, placing the common denomi- 
nator 9 under 25, we have che improper fraction 2°, or 2 
whole numbers or integers, and 7, making the mixed 
fraction pa 

To add and subtract fractions not having a common | 
denominator, bring the different denominators to a com- 
mon multiple, or the least common denominator, and raise 
the different numerators by the common denominator in 
the same proportion, after which add as above. 

To bring different fractions. to a common denominator, 
the largest. denominator is retained, but all other denomi- 
nators which, according to the properties of figures and 
numbers, can be resolved into any common factor con- 
tained in the largest denominator or any other denomina- 
tor which is retained, are cancelled or thrown out of the 
question, after which multiply the remaining denomina- 
tors or figures or numbers to a continued product, for the 
least common denominator. 


Example. Find the least common denominator or mul- 
tiple in the following series of denominators, from 2 to 10 
inclusive, viz., 2, 3, 4, 5; 6, 7, 8,9, 10; 10 being the largest 
denominator, is retained; 9 hasno factor in common with 
10, therefore 9 is retained: Shas a relation or factor in 
common with 10, viz., 2, which cancelled into 8 leaves 4; 
therefore 4 is set down: 7 has no relation or factor in com- 
mon with 10, 9, or 4; therefore 7 is retained: 6 has two 
factors, 2 and 3; the two is contained or cancelled in 10, 
and the 3in 9; ‘therefore 6 is left out: 5 is contained or 
cancelled in 10, therefore 5 is left out: 4 is contained or. 
cancelled in 4, therefore 4 is left out: 3 is contained or 
cancelled in 9, therefore 3 is left out: 2 is contained or 
cancelled in 4 or 10, therefore 2 is left out. 

Now, the numbers retained or remaining, viz., 10, 9,4, 


62 ADDITION AND SUBTRACTION OF FRACTIONS. 


7, multiplied to a continued product, form the least com- 
mon denominator of the above numbers. viz., 2520, by 
which raise the numerators in the same proportion ; that 
is, by dividing the common multiple by each denominator, 
and multiplying the quotient by the numerator of the re- 
spective denominators ; and this done, then add as pre-~ 
viously. 


| EXAMPLE, 

ge Atte, Sie ee. 840—-10=, 84 

! 840 S=105 

at, Mie ine 8402. 7=120 

_ 10 10 » . $401, .7=120 

echily 4 840° 21== 40 
SRT aan Tithar oO 

— 7 §40 Common denom. 2351 
23518402671 Answer. 


But to return to our principle of cancelling again. Sup- 


pose it be required to divide the product of 144, 77, 49, 


24 and 96 by the product of 9, 16, 11, 13, 3, 6 and 7. 

The statement to the above example, and the follow- 
ing, will be solved without repetition, that the learner 
may obtain accurate views relative to the mode of solution 
here presented. 

Statement ta the above example : 


—9 | 144— 
THI Gs| ole 
—li 39— 
—13 24— 4 

—3 96 

tis 

eee Tolh 

| 384 


The numbers marked are cancelled, those remaining 
unmarked are 4 and 96, being multiplied together give the 
answer, 384. The result would have been the same had 
the right hand numbers been multiplied continually to- 
gether, and divided by the continued product of the left 
hand figures and numbers. 


ee 


ADDITION AND SUBTRACTION OF FRACTIONS, 63 


Divide the product of 99, 49, 15, 90832; 13, 16 by the 
product of 77, 10, 16, 49, 39 and 12. 


7 —77 | 99— 9— 3— 
—10 | 49 
—16 | 15 
—49 | 2 0— 
—3 —59 ; 32— 8 
—4 —12 | 13— 
| 16— 


7 | 240—342 Answer. - 


In the solution of the last example, we first observe we 
have two 49s, one on each side of the line; these we can- 
cel; also eee 16s, these we strike out lee we have also 
two ciphers, these we cancel also; now, 77 and 99 havea 
common factor of 11, we cancel them both and substitute 
their quotients; 13 is also contained in 39, 3 times, we 
cancel both numbers, and place the 3 on the side of the 

reater number; this 3'is also contained in the 9 on the 
right hand side of the line 3 times; these we cancel, and 
place the 3 on the side of the greater; then this 3 is con- 
tained in 12 on the left 4 times, cancel the 3 and 12, and 
place the 4 on the side of the greater number; now, 4 is 
also contained in 32 on the right 8 times, these we also 
cancel and place the 8 on the side of the greater number. 
Now, ‘it will be observed that we can cancel no farther, 
since there are no two numbers that will divide without a 
remainder :+we have remaining on the right_15, 2 and 8, 
whose continued product is 240; this being divided by 
the remaining number on the left, viz., 7, gives the quo- 
tient 342, the answer required. 

Again: Suppose it is required to multiply 8001 by 735, 
this product to be divided by 7, this. quotient to be:multi- 
plied by 33 times 51, this to: be divided by 11 times 2667, 
this quotient again to be multiplied by 84 times 50, aa 
finally to be divided by 34 times 81 for the answer. To 
perform the operation in the usual way it would require 
the following operations : 


64 ADDITION AND SUBTRACTION OF FRACTIONS. 


To multiply 8001 
by | 735 


2d. To be divided by 7)5880735(840105 
33 "P1683 
51 ——— 
os nin 520315 
33 6720840 
165 | 5040630 
683 ra nda 
11-+-2667=29337)1413896715(48195 
117348 
240416 
234696 
57207 
29337 
278701 
, 264633 


146685 
| 146685 
And to multiply 48195 by 844504200 
4200 


9639000 
192780 


To divide by) 
34-181=2754) 202419000(73500 
19278 
9639 
8262 
13770 
13770-——-—00 


MULTIPLICATION OF FRACTIONS, 65 


But according to our simplified system we would pro- 
ceed thus: 


SEU Uk eam 
_  —I1 | 735 
2667 | i lef ala 
orga Th Py 
ge ot Sigh real foe. ws 9 
| 50 


(RR Ree ee ena ee ee NY ce ee, ee Se 


| 73500 Result. 


By the usual process it requires 235 figures. 
Our simplified process 4 
Difference 193 


MULTIPLICATION OF FRACTIONS. 


Rule. Place the numerators, both of the multiplicand 
and multiplier, on the right hand side of the perpendicular 
line, and the denominators on the opposite side. 

Note. The reason for thus placing the numerators on 
the right, and the denominators on the left, is, that nu- 
merators are dividends, and denominators are divisors. 


EXAMPLE, 


Multiply 2 of t of 4 of +3 by 2 of $ of 48 


Feit sty sa 
—60 | 49— 
9 — 1—3 Answer. 


66 MULTIPLICATION OF FRACTIONS. 


a7) Required the cost in dollars and cents of 2 3 OF ro “4 


yt off 
2 ofa yard af pilot gti: at 4 of & 5 of Fy of 2 
3 of 33 of =3, of a dollar per Jon. 
Multiply 4 of 3 of 7 of 5 of 18 by 4 
of 40. 


mwWOnG eR OO 
(o2) 


| | 


—2 


Lalit? J 


ont ODN 


| 
bo AIS ie 


jt 


| 15 Answer. 
_. Multiply $ of 4 of £.of & of s& of 44 by 4 
1000. 


{ 24 Answer. 
Multiply 34 of 2 
9) ae 20+ A 
ie 
15— 
—10 | 
—24 | 


73— 
8 


BB 


192541 Answer. 


2 of 12 of 43 of 33 ete of 2 of 2 of 8 of of 3 43 
‘at 31 “of 2 oF 
ee $ 0. 00. 166-4. 

of 51, of = of fy 


5 of +2 vos of 


ib ohgeh of 15,by 735 of . of $ of s%z. 


(or) 
ae 


MULTIPLICATION OF FRACTIONS. 


FT; Multiply Eby +. Ans. + 
a a 1 by . Ans. + 
2a: 5». .apor 2 by = Ans. 3 
4 a 4 by dof 3 _ Ans. 4 
5 4 of 2 of by 4 of 3 Ans. 3¢ 
6. “ t of 9, of $ of & by 2 of & Ans. 4 
7. ro 2 of 2 of 2 of 4 by $ of # of F Ans. 4. 
8 . Sof 2 of by + of 8 : Ans. 2 
9. - Lof 8 of 4 of 2 of 5 by.2-0f 3 Ans..1 
10. ai Sof ¢ of 4 of L by 3 of 5 of 4 of 10. Ans. §. 
Hf, = 4 of € of 34 by + of 2 of 5. Ans. 4. 
12. y 2 of £ of + of 44 by 4 of 34 0f 54. Ans. 1 
13. «2h of S of 8 by 34 0f 4 of Sof 4. Ans, 96 
14. 3 of t of 4 of 35 of by 44 of 55. Ans. 3,4 
15. «AS by ¢§ of 4. Ans. I. 
16. «34 of 2 by “at of 4, Ans. 1. 
ee « 52 of $ of 3 by Lof 8. Ans. 44, 


18. «6-28 of ~ of 2 of Eby 4 and $ of 3. Ans. 4. 
19. What will 24 ie. of sada cust at 14 ec net lb. 2? 


Ans. 24 cts. 
20. Required the cost of 34 Ibs. of pork at 44 cents 
per lb. Ans. 16 cts. 
21. Required the cost of + of 4 of 34 yards of tape at 
2} cents per yard. Ans. 14 ets. 
22, What will # of 2 of 2 of 24 yards of ribbon cost at 2 
of 52 cents per yard 2 a Ans. 9 ets, 


24. A gentleman purchased £ of 3 of 4+ of 7 wae of 
cassimere at 2 of 32 dollars per ‘yard, Required the cost. 
Ans. + of a dollar. 
25. Required the cost of 2 z of £ of # of $ OF d4 yards of 
satinet at 4 of 3 of 32 déllars per yard. 


Ans. 24 dollars. 
26. Required the cost in dollars and cents of 4 1 of 3 of 
$ of 4 OE af e . of £ of 42 yards of broadcloth at § of 35 of 
3 of h ii ae OF $2 “dottars yer yard. Ans. $1 "3142: 
“AW hate vai 4 of 8 of 2 baat dae of 8 of 22 maeds. cloth 
in dollars nd Sante, sit + a 13 “of 4 of 121 dollars per 
yard ? Ans $2 50. 


68 = DIVISION OF FRACTIONS, © 


28. A igertlenaty purchased 4 of =3, of 2 of § of 3 of 42 
of = of tof a yard of broade ‘loth at 3 off 5 of i of 8 dol. 
lars per yard. Heqesren the cost in dollars and cents, 

Ans. $0 18 61-+-. 


DIVISION OF FRACTIONS. 

Rule. Place the numerators of the dividend, and the 
denominators of the divisor on the right hand side of the 
line, and the denominators of the dividend and the nu- 
merators of the divisor on the left hand side. 

Note. This is called inverting the divisor. 


EXAMPLES. 
Divide } of 7 of + of 13 of 23 by ¢ of +5 of 9, of 3. 
| es 
: | i 
cay Aa (pe nts 
—13 | 12— 
—4 —48 | 39-—— 3 
bo 
a 
BIGUP 1 Aes A 
25 | Ones 


# 


16. | 27=111 Answer. | 
Divide 35 of +3 of 7 of 43 by } of 3 of 3, of 3 of 100. 


L219] 9 

pUn penny Ny heen 
oT Ga 
Meo oT 
eA | 6— 
aS Gea dee 
= | 12 
— 63) 20— 

25 —100 | 


a ee ee 


125, |. 108S=1$8 Answer. - 


MULTIPLICATION: AND DIVISION OF FRACTIONS. 69 


Divide 3 of 3; of } of = of 44 by $3 of 4 Sten of 60, 


—5 | 3— 
4 —12 | 6— 
=f | 6. 
7 | o— 
—d5 | 24— 
ane ee | 5,0— 
1 | 2— 
—5 | S—_ 
4 | 5— 
60 | 
112 | 5= 35 Answer. 
Divide =; of 7°, of 12,3; by 8} of 46, of 10. Ans. qe: 
Divide 3+ of 54, of of 60 by 3 of 2 of 44 of 39. 
ahs Ans. 128 
Divide # of 1 of ;3, of 2° by 3 of 2 EOF TEP. Ans. 3, 
Divide 100 by 4 of $4 of $ oo of 49 of 60, Ans. 1722, 
80 


Divide 2 of Z of 80 by Sof , of 2° of 8, of 444. 
Ans. 143 
18 men purchased 3 of ? of 2 of { of 24 yards of ae 
and divided it equally, Required the share of each. 
Ans. 35 ofa yard. 
4 men bought 4 of 3 of 2 2 of 44 pounds of sugar, and di- 
vided it equally among them, Required the sali of each 
TaD Ans. |, of alb. 
12 men purchased + of 2 of § of 4 of 3 of & of 10 of the 
a Hey By: What part d ‘did each. own? Ans. 235 


432° 


MULTIPLICATION AND DIVISION « OF 


FRACTIONS. 
1. Divide ¢ of 19 by 3 of 2, and-multiply it by 4 of 6. 
Bisco Bh ae 
19 

ig jae 

met a 
i 
Gre Fo 


aT ee ey See 


| 19 Ans. 


70 MULTIPLICATION AND DIVISION OF FRACTIONS. 


© 2) Divide 34 of % by 4 of 10. Multiply it by 3.of 9. 
Divide the product by 7 of 3.. Multiply again by 23 of 7 
for the answer. 

10— 
Dy gacoss 
G— 


—3 
—6 


i 


—10 
—Ss 


9— 3— 
eb 
—3 
—sd | 12 

| 7 Beas 


5) 


| 60 Ans. 


8 Divide 2 of 2 by 4 of 8. Multiply it by 8 of 3 of 12. 
Divide by $ of 12. Multiply by 3 of 20. Divide'by 5%; 
of 8. Multiply by ; of 30. | | 


—3 | 2 
ise was : —4 6 
8 
—3 —9 | 8— 
7 | 3— 
12— 
—12 | 3— 
—d | 20— 5— 
—5 |} 12— 2 
—Sl. 
—6 | 4— 
: | suas 6 


a 


7 | 24=33 Ans. 
4. Divide 202 by §}. Multiply by 302. Divide by 58. 


MULTIPLICATION AND DIVISION OF FRACTIONS. 


Multiply by 4 of 7. Divide 
100. 
—5 


—d5l 


3 —12 


yi 


by 103. Multiply by ~% of 


a a a 


3 | 5600=18662 


5. Divide 4 of 2 by 2of 3. 
vide by } of g of ob 


—8 
—2 
—3 

= | 


—9 


6. Divide 24 by 43, of 12. 
60. Divide by % of 360, 
—30 


—12 


— 
ePigteal 
8 


3 


Q wd ---20 ---360 


Gorecey EEN EE CER 


Ans. 


Multiply by 4 of 8 of 2. Di- 


Multiply by 23 of 360. 
4— 


3 
y ee 
6— 2 


64 Ans. ) 
Multiply by % of tof § of 
Q4-.- 2... 

18--- 


tae 
p ale 
By oatas 
| 60+ 2-- 
Qeax B--- 


6 | l=. Ans, 


79 REDUCTION OF FRACTIONS. 


7. Multiply 4 of 8 of 9. Divide by * of 5. Multiply 
by 3 of 6. Divide by $+ of 15. Multiply by $ of 12. Di- 
vide by @ of 6. Multiply by $$ of 4. Divide by = of 12. 
Multiply by 2¢ of 100. 


REDUCTION OF FRACTIONS. 


Rule. Place the numerator of the given fraction on the 
right hand side of the perpendicular line, and its denomi- 
nator on the left, then place also on the left, such numbers 
as are necessary to reduce the denomination given to that 
required, then proceed as before. 


EXAMPLES. 


Reduce # of a farthing to the fraction of a shilling. 
~~ 13 far. 
far 4 | 1 penny 
pence 4 --12 { 1 shilling 


~“64 | 1 er part ofa shill: Ans: 


REDUCTION OF FRACTIONS, 73 


2, Reduce $ of a penny to the fraction of a pound. 


300 | 1st y Ans. 


3. Reduce 1% of a gallon to the fraction of a hogshead. 
10 | 7— 
9 —63 | 


te ree 


90 l= Ans. 


4, Reduce 3 of one ounce Troy to the fraction of a 


pound. Ans. 6. 
5. Reduce *! of a pound, avoirdupois, to the fraction of 
a cwt. ANnSi-y# 4 « 
6. Reduce 5 of anail to the fraction ofan Ell French. 
Ans. 64. 
7. Reduce * of a BPeUEY, to the fraction of a pound, 
Ans. 5 , Gis 


8. Reduce {2 of an hour fo the fraction of a year. 
9 Ans. 94'5 0. 
To reduce fractions of high denominations to equivalent 
fractions of low denominations. 


Rule. Place the numerator of the given fraction on the 
right hand side of the perpendicular line, and the denomi- 
hater on the left as before, then. place on the right hand 
side of the line such numbers as are necessary to reduce 
the denomination given to that required, then proceed as 


before. 


EXAMPLES. 


9, Reduce ss of a shilling to the fraction of a farthing. 
2—8 —96 | 1 
| 12— 
idee 


2 | l=} ofa farthing. Ans, 


74 REDUCTION OF CURRENCIES. 


10. Reduce ,}, of a pound to the fraction of a penny. 
3 —360 | 1 
| 2— 0 
| 12— 
3 | eee Ans. 
11. Reduce +; of a pound Troy to the fraction of an 
ounce. 


§ —15'|1 
(12— 4 
5 | 4=4 Ans. 
12. Reduce ;}, of a penny to the fraction of a farthing. 
Ans, 4. 
13. Reduce 9 ¢oy00 Of a mile to the fraction of a bar-, 
ley corn. Ans. 2°... 
14. Reduce +35 of an Ell English to the practice of a 
nail. Ans, 3. 
15. Reduce 57455 of a ba to the fraction of an hour. 
Ans. + 


REDUCTION OF CURRENCIES. 
UNITED STATES MONEY. 


The New England } 6 ts equal $1,00 
Virginia a 2000 


Kentucky Ls 3 shillings aig 50 
‘Tennesse 418 penee * 25 
8 shillings equal $1,00 
New York ie a «5.00 
North Carolina { shillings “ 20 


4.24 pence: (, #4, 125 
Pennsylvania q 7 shill. 6 pence equal $1,00 


New Jersey BS « 800 
Delaware 3 shillings “ AO 
Maryland 5 9 pence és 10 
South Carolina ! ea 8 pence pie 
G ° by shillings 1,50 
ae 4 14 pence 525 


REDUCTION OF CURRENCIES. yi 


T'o change pounds shillings and pence into dollars and cents. 


Rule. Place the number of pounds given, on the right 
hand side of the perpendicular line, then see the propor- 
tion of United States’ money, and place the number of 
pounds in that currency on the left hand side, and the 
equality of dollars on the right. 

Prop. 1. Reduce £160, New York currency, to federal 


money. | Ans. $400. 
£--2 | 160—80 
1 5$ 
400$ Ans. 
2. Reduce £240, New Jersey currency, to federal 
money. Ans. $640. 
3. Reduce £243, New Jersey currency, to federal 
money. $648. 
4, Reduce £140, South Carolina currency, to federal 
money. . $600. 
5. Reduce £560, South Carolina currency, to federal 
money. $2400. 
6. Reduce £27, New England currency, to federal 
money. $90. 
7. Reduce €80, New York currency, to federal money. 
$200. 
8. Reduce £45 12s. New Jersey currency, to federal 
money. $121 60. 
9. Reduce £112, Georgia currency, to federal money.. 
$480. © 


To change dollars and cents to pounds, shillings and pence. 


Rule. Place the number cf dollars given, on the right 
hand side of the perpendicular line, then see the propor- 
tion of United States money, and place the number of 
dollars in that currency on the left hand side, and the 
equality of pounds on the right. 


76 SIMPLE PROPORTION. 


Prop. 1. Reduce $648 to pounds, New Jersey currency. 


Ans 243 pounds. 
+-§ | 648—S1 
Se 
£243 Ans, 


2. Reduce $450 to pounds, New York currency. 
Ans. 180 pounds. 
3. Reduce $360 to pounds, New England currency. 
Ans. 108 pounds. 
4, Reduce $240 to pounds, South Carolina curreney. 
Ans. 56 pounds. 
5. Reduce $580 to pounds, Canada currency. 
Ans. 145 pounds. 
6. Reduce $642 875 to pounds, south Carolina cur- 
rency. Ans. 150 pounds. 
_7. Reduce $141 to pounds, New England currency. 
. Ans. 42£ 6s. 
8, Reduce $250 to pounds, Canada currency. 
Ans. 62£ 10s. 
9. Reduce $125 60 to pounds, New Jersey currency. 
Ans. 47£ 2s. 
10. Reduce $475 75 to pounds, New York currency. 
Ans. 190€ 6s. 
‘11. Reduce $75 to pounds, New England currency. 
| Ans. £22 10s. 
12. Reduce $384 to pounds, Nova Scotia currency. 
Ans. 96£. 


SIMPLE PROPORTION. 


In this rule there are three terms given to find a fourth 
that shall have the same proportion to the demand, that 
the denomination corresponding with the answer does to 
the denomination corresponding with the demand. The 
demand may frequently be known by questions like 
the following, viz: How much? How many? What 
cost? What will, &e. : 


Rule, In stating, notice whether the demand and its cor- 
responding term be of the same denomination, if so, place 
the demand first on the right hand side of the line, and its 
corresponding term immediately opposite on the left, and 


SIMPLE PROPORTION. 77 


the remaining term which must be of the same denomina- 
tion with the answer on the right hand side and the last 
mentioned. But, if the demand and its corresponding 
term be not of the same denomination, and the remaining 
term be not of the same denomination of the required an- 
swer, they may be reduced to the same denomination by 
the following rule : 

Iirst, write the demand on the right hand side of the line, 
then write the numbers necessary to reduce that quantity 
to the denomination for which the price is given on the 
right hand side of the line under each other, together 
with the price, then on the left hand side of the line write 
the numbers necessary to reduce the said price to the de- 
nomination of the reqtired answer. 


Note. Notice which of the given terms is of the same 
kind or name as the required answer, and place it on the 
right hand side of the line. Notice again whether the re- 
quired term must be greater or less than this, and if great- 
er, place the greater of the two remaining terms under 
the preceding term, and also on the right ar the line, and 
the less of the two terms on the left, but if less, place the 
less of the remaining terms on the right of the line, and the 
greater on the left, then proceed as befovre. 


EXAMPLES. 
1. If 3 yards of cloth cost 7 dollars, how many dollars 
will 9 yards cost ? 


yds. yds. demand 
corresponding with —3[ 9— 3 


the demand. $7 corresponding with the 
answer. 
| $21 Ans. 
2. If 4 Ibs of sugar cost 35 cts., what will 12 Ibs cost. 
—4]12— °3 
= 30 


eres ee 


| $1,05 Ans 


78 SIMPLE PROPORTION. 


3. If 8 men can mow l5acres of grass in a given time, 
how many acres will 48 men mow in the same time? 
—8 ue 48— 6 


| 90 acres. Ans. 


4, If 9 yards of cloth cost 63 dollars, how many dol- 
lars will 45 yards cost ? 
—9 
is 7 


$315 aye 


5. If 48 men can build a wall in 24 days, how many 
men can do the same in 192 days. 


—s8 —192 | 24— 


48— 6 


| 6 men. Ans. 


6. If 160 poles long and I pole wide make an acre, 
how much in length, that is 8 poles wide, must be taken to 
contain anacre 4 


—8 }1 
160— 20 
[20 Ans. 


7. How many men must be employed to finish a piece 
of work in 15 days, which 5 can do in 24 days? 
—3 —15|24— 8 
fits 


| 8 men. Ans. 


8. Ifa man perform a journey in 6 days, when the day 
is 8 hours long, in how many days will he do it, when the 
day is 12 hours long? 

—2 —12|8— 4 
oe 


| 4 days. Ans. 


SIMPLE PROPORTION. 719 


9. If I lend my friend $100 for 180 days, how long . 
ought he to lend me $450 to return my kindness ? 
—d —450 | 100— 20 
‘| 180— 2 


| 40 days. Ans. 


Note. Should there fractions occur in any of the exam- 
ples under this rule, the numerators should be used like 
whole numbers, that is, the numerators of the 2d and 3d 
terms should be placed on the right hand side of the lines, 
and the numerators of the Ist term on the left hand side 
as in division of fractions, the denominators are always 
placed opposite their own numerators. 


10. If 6 lbs. of iron cost ny cts., what will 3 of a lb. cost ? 


alg aly eae i19 


a 


7° | S0=-67- cts. | Ans. 


11. A person having 3 # of a coal mine, sells ? of his 
share for $171, what is the whole valued at? | 


—3 {4 
bat AB: 
171— 19 
| $380 Ans. ~ 


12. If 73 yds. of cloth cost 24 dollars, what will 30 yds. 
cost ? 


—3 —15 | 30— 10 
yea 
29 fin 

: $10 Ans. 


13. If 4% barrels of flour cost $25, what will 37 barrels 
cost ? 
---37 | 37--- 
8 
25 


(ee pes ee 


: $200 Ans. 


80 SIMPLE PROPORTION. 


~- 


14. If +. of * ofa yard of cloth cost 60 cts., how 
much will + of # of 4 yards cost ? 


wep ilg) OF 
ba} Tapas 
Ss we 
hal) 5. 
«m5 bs] Dom 
60--- 3 


ee are ee 


| 72 cents. Ans. 


15. If 24 pounds of tobacco cost 26 cents, what. will ¢ 
of 4 of 9 pounds cost ? 


ads 0 
Mae Pe 
9 
£413) eB 
rate 2 


| 54 cents, Ans. 


16. If 4 of 2 of # of $ of a yard of linen cost 18 cents, 
what will 3 yards cost! 


3 
1 | 2--- 
291 9.5 
meesereitre 
.-4 |-9 
18 
| $4,86 Ans. 


17. If 34 times 34 yards of cloth cost 14 times 14-£s, 
what is the value of 3 of } of 124 yards? 
1 


mtg 
Pest T 
---4 | 49--- 
nonfat dene 
en AO. 
A 19... 

Bis 

20--- 5 


2! 157% shillings. Ans. 


ee . 


SIMPLE PROPORTION. 81 


18. If 1 pint of wine cost 10 pence, how many &s will 
3 hhds. cost ? 


hhds. «1 | 3---.  hhds. 
gall. 1/63 ~ gall. 
qts. 1 | 4--- qts. 
pts. 1 | 2--- pts. 
pence ---3----12 | 10--- pence 
shildy :- ---20 | 1 shill. 
fl abe 
| 63." Ans. 


Nore. In the above example the demand is laid on 3 
hhds., this we place first on the right hand side of the line, 
this we reduce to pints by placing the several numbers 
requisite to reduce hhds. to pints. (See the table of wine 
measure.) After we have gone to pints, reduction de- 
scending on the right, we then place pints opposite on the 
left, and the price on the right hand; we then proceed to 
reduce pence to pounds, the denomination required on 
the left hand, (see pence table) and the denomination £s 
falls on the right hand side ofthe line. The name or de- 
nomination of which we wish our answer, must always be 
the last on the right. 


19. If 3 pts. port wine cost 9 pence (New England cur- 
rency,) how many dollars will 3 hhds. cost? (See table.) 


ee 
63 
7 a 
val | Sane 
12 | 9- 
aA 
| $63 Ans. 


Nore. It is not necessary to use the figure 1’s, for they 
are not regarded in the calculation, therefore, they are 
understood and not expressed. 


8* 


82 SIMPLE PROPORTION. 


20, If 2 qts. of cider cost 1 shilling (New York,) how 
many dollars will 4 hhds. cost ? 


she 
63 

£28 4 4 

---8 
} $63 Ans, 


21. If 6 gills cost 3 farthings (New England) how many 
‘doliars will 12 hkds. cost? 


12--- 
63--- 21 
ets 
2 

AEC leet 

éscbche Been 

---3 ---12 

---6 

($42 Ans. 


22, If 3 qts. of oil cost 6 shillings (New York,) how 
many dollars will 2 tuns cost ? 


vw 


Done 
Oe ae 
| 63 
4-844 

‘61 $2038 


' $504 Ans. 


23. If 4 lbs. of tobacco cust 2 shillings(New York,) how 
many dollars will 8 cwt. cost ? 


i 8 
AD 
---4 | 28 
i 2 
1 $56 Ans. 


SIMPLE PROPORTION. 


24. If 9 lbs. of nails cost 6 shillings (New York,) how 

many dollars will 30 tons cost? 

. 3825 

Aoi 
ao 
---3 ---9 | 28 

sae t 6D 


' $5600 Ans. 


25. If 4 quarts of oats cost 16 pence, how many dollars 
will 60 bushels cost, New Jersey currency? Ans. $852. 

26. If 3 pecks of beans cost 7 pence, how many dollars 
will 9 bushels cost, South Carolina currency t Ans. $14. 

27. If 12 drams of opium cost 30 pence, how many del- 
lars will 6 pounds cost, New York currency? Ans. $15. 

28. If 12 grains of silver be worth | shilling, what is the 
value of a silver tankard, weighing 4 pounds ? 


Ans. $256. 

29. Bought a piece of cloth for £164, at 15 shillings per 
yard, how many yards did it contain ? Ans. 22 yds, 
30, If 14 yards of cloth cost 24 dollars, how many cents 
cost 14 quarter of a yard ? Ans. 624 cts. 
31. If 2 of a yard cost ;°5 of a dollar, how many cents 
cost ,5 of a yard? Ans. 21 cts. 
32. When 19 pounds of sugar cost {% ofa £, how many 
ounds can I have for 3 of a shilling 4 Ans, 2 lb. 


33. If 84 ofa lb. of starch cost 3 of a £, how many 
pounds of starch can I have for 72 cents, New Jersey 
currency? Ans. 6 lbs. 

34. If 2 of a yard of cloth be worth 4 of two dollars 
and 28 cents, what is the value of 7 yards? 

Ans. $17,738! cts: 

35. If4 ofa yard of cloth, = wide, cost 24 dollars, what 


is the value of 24 yards, 14 wide. Ans. $221. 
36. If ? of 4 of the cargo of a ship be worth 250 dollars, 
what is the value of the whole cargo 4 Ans. $1333, 


37. If 4 pounds of nails cost 18 pence, how many dol- 
lars will 12 tons cost, New York currency? Ans. $1260. 
38. If 3 of ¢ of ¢ of aship be worth § of $ of 11 of the 
cargo valued at 12000 dollars, what did both ship and car- 
go stand the owner in 4 Ans. $15223,442°7'3 CtSen 


84 SIMPLE PROPORTION. 


39. If 4 of 8 of f of ,%, of aship be worth % of 7 of 4 
of 8 of the cargo valued at $15000, what did both ship and 


cargo stand the owner in? - Ans. $23000. 
40. If 12 men build a house in 48 days, in what time 
could 36 men build it ! Ans. !6. 


41. Admit that I lend a friend on his occasion 100 dol- 
Jars for six mouths, and he promised me the like kindness 
when I desired it; but, when I come to request it, he 
could lend me only 75 dollars. ‘The question is, how long 
must I keep the 75 dollars to recompense my courtesy to 
him ? / 

After the direct pure proportion, the demand would be 
laid upon 75 dollars; but we invert or change, and lay the 
demand upon the 100. Ans. 8 mo. 

* 42, If 1 lend my friend 100 dollars for 6 montas, allow- 
ing the month to be 30 days, how many days ought he to 
lend me 1000 dollars? * Ans. 18 days. 

43. If, for 48 shillings, 225 cwt. be carried 512 miles, 
how many cwt. may be carried 64 miles for the same 
money ? Ans. 1800 ewt. 

44, If, when wheat is 83 cents per bushel, the cent loaf 
weighs 9 oz., what ought it to weigh when wheat is 1 dol- 
lar 245 cents per bushei 4 Ans. 6 oz. 

45. There is a cistern having a cock which will empty 
itin 12 hours, how many cocks of the same capacity will 
empty it in 4 of an hour ? Ans. 48. 

46. A gentleman purchased 24 yards of cloth, at 3 shil- 
lings per yard. Required the cost in dollars and cents, 
New York currency. Ans. $9. 

47. Purchased 36 yards of satin at 4s. 6d. per yard. 
Required the cost.in dollars and cents, New England cur- 
rency. Ans. $27. 

48. A gentleman bought 108 pounds of tea, at 7s. 6d. 
per pound. What was the cost in dollars and cents, New 
Jersey currency ? Ans. $108. 

49. Required the cost of 28 pounds of young hyson 
tea, at 6s. 8d. per pound, in dollars and cents, South Caro- 
lina currency ? Ans. $40. 

50. A man bought 3 hogsheads of molasses, each hogs- 
head contained 120 gallons, at 1s. 8d. per gallon. Re- 
quired the cost in federal money, New York currency. 


Ans. $75. 


SIMPLE PROPORTION. 85 


51. A gentleman purchased 8 hogsheads of oil, each 
hogshead contained 140 gallons, dae 6s. 2d. per gallon, 
Required the cost in federal money, New England cur- 
rency. ‘Ans: $11514. 

52. Purchased 4 pieces of cloth, each piece containing 
30 yards, at 1s. 4d. per yard. Required the cost in fede- 
ral money, New Jersey currency. Ans. $214. 

53. A gentleman bought 2 bales of cloth, each bale con- 
tained 42 pieces, and each piece 30 yar ds, at 2s. 6d. per 
yard. Required the cost in federal money, South Caroli- 
na currency. Ans. $1350. 

54. A gentleman purchased 12 hogsheads of molasses, 
each hogshead contained 120 gallons, at 3s, 4d. per gallon, 
and paid for the same with cloth, at 6s. 8d. per yard, Re- 
quired the number of yards he gave in exchange. 

Ans. 720 yards. 

55. A merchant bought 12 tons of iron, at 4d. per 
pound, and paid for the same with molasses, for which he 
was allowed 2s. 4d. per gallon. Required the number of 
hogsheads he gave in exchange, allowing each pe 
to contin 80 gallons. Ans. 

56. A gentleman purchased 6 tons of bar iron, ‘at ah 
per pound, (New York currency,) and paid for the same 
with candles, for which he was allowed 15s., (New Eng- 
land currency,) per box. How many boxes of candles 
were required ? Ans. 336. 

57. A philanthropist distributed a certain amount of 
money among 42 poor widows, giving them each 4s. 2d. 
Required the amount of his distribution in federal money, 
South Carolina currency. Ans, $37,50. 

58. Supposing the circumference of a wheel to be 15 feet, 
how-many times will it revolve in going from Boston to 
Dedham, the distance being 10 miles?) Ans 3520 times. 

59. How many times did Captain Cook sail the length 
of his vessel in circumnavigating the globe, the circum- 
ference being 24,800 miles, and the length of the vessel 

240 feet ? Ans. 545600 times. 
‘ . The circumference of a large wheel is 36 feet, and 
that of a small one is 18 inches. How many more revolu- 
tions will the latter make than the former, in going from 
Schenectady to Rochester, the distance being 140 miles ? 

Ans. 472 266+: 


S86 SIMPLE PROPORTION, 


61. The distance from Lowell to Boston is 26 miles, 
allowing the average width of the road to be 4 rods, how 
many ucres would be contained there? Ans. 208 acres. 

62. If 3 yards of cloth cost 7 dollars, how many dollars 
will 9 yards cost ? Ans. $21. 

63. What will 12 pounds of sugar cost if 4 pounds cost 
35 cents? Ans. $1,05 cts. 

64. If 3 yards of cloth cost 15 dollars, how many dol- 
lars will 12 yards cost? Ans. $60. 

65. If 8 men mow 15 acres of grass in 3 days, how 
many acres will 48 men mow, in the same length of time ? 

Ans. 90 acres. 


66, If 4 yards of cloth cost $8, what will 26 yards cost ? 


Ans. $52. 
67. If 2 yards of muslin cost 46 cents, what will 8 yards 
cost ? Ans. $1 84. 


68. If 7 horses consume 21 bushels of oats in 3 days, 
how many bushels will 3 horses consume in the same 


time? Ans. 9 bush. 
69. If 28 lbs. of butter cost $5 88, what will 7 lbs. 
cost ? Ans. $1 47. 
70. If 3 yards of cloth cost $9, how many yards will 
$243 buy? Ans. 81 yards. 
71. What will 30 lbs. of sugar cost, when 45 cents will 
buy 5 lbs. Ans. $2 70. 
72. lf 20 yards cost $120, how many yards may I have 
for $30 4 | Ans. 5 yards. 
73. If 7 lbs. of sugar cost 56 cents, how much will $7 12 
buy ? Ans. 89 lbs. 
74. If 3 cords of wood cost $4 35, what will 27 cords 
cost? Ans. $39 15. 
75, If 4 yards of cloth cost $35 50, how many yards 
may be bought for $106 50 ? Ans. 12 yards. 


76, If 12 yards cost $9 72, what will 192 yards cost ? 
Ans. $155 52. 


77. How many bushels of wheat can be bought for $24, 


when 6 bushels cost $9 2? Ans. 16 bush. 
78. If 7 lbs. of sugar cost 63 cts., what will 25 lbs. 
cost 2 Ans. $2 25. 


79. If aman can travel 15 miles in 3 hours, how many 
miles can he travel in 5 hours 2 Ans, 25 miles. 


SIMPLE PROPORTION. 87 


$0. How many laborers must be employed to finish a 

piece of work in 15 days, which 5 men can do in 24 days? 

Ans. 8 men. 

81. If 12 men can build a house in 30 days, how many 

will do it in 8 days? Ans. 45 men. 

82. If a man perform a journey in 6 days, when the day 

is 8 hours long, in what time can he do it when the day is 

12 hours long ? Ans. 4 days. 

83. If lL lend my friend $100 for 180 days, how long 
ought he to lend me $450 to return my kindness? 

Ans. 40 days. 

84. If 13 men can perform a piece of work in 35 days, 

in how many sae would 5 men perform the same work ? 


Ans. 91 days. 
85. If 7 men do a piece of work in 16 days, how many 
men can de it in 4 days ? Ans. 28 men. 


86. If 20 horses eat 35 bushels of oats in a week, how 
many bushels will 8 horses eat in the same time ! 

Ans. 14 bushels, 

87. If 20 men can mow a field in 34 days, how many 


men*can mow it in 8 days? _ Ans. 85. 
88. [f 8 men can build a wallin 20 days, how long will 
ittake 5 mento build it? .” Ans 32 days, 


89. If 20 men can perform a piece of work in 35 
days, how many men can do it in 7 days ? 
‘Ans. 100 men. 
90. If 12 oxen can eat 5 acres of grass ina week, how 
many acres will it take to keep 36 oxen the same time 1 
Ans. 15 acres. 
91. If my friend lends me $300 for 36 days, how long 
should I lend him $80 to repay his kindness } 
Ans. 135 days. 
92. Suppose aman paints a house in 45 days, and 
works 8 hours a day, how long would it take bim if he 
worked 9 hours a day? Ans. 40 days. 
93. A man borrows of his friend $280, which he keeps 
40 days; how much must he lend his friend 70 days as an 
equivalent ? Ans. $160. 
94. It takes 84 yards of paper that is 32 inches wide, 
to cover the walls of aroom; how many yar ds will it take 
to cover another room of the same size, when the paper is 
24 inches wide? : Ans. 112 yards. 


838 SIMPLE PROPORTION. 


95. How much in length, 44 inches broad, will make a 

foot square 4 Ans, 32 inches. 

96. There is a cistern, having a pipe which will empty 

itin 15 hours; how many pipes of the same capacity will 

‘empty it in 3 quarters of an hour 4 Ans. 20 pipes. 

97. What is the height of a tree, whose shadow is 180 
feet, when a staff 5 feet long casts a shadow 9 feet 4 

; 7 Ans. 100 feet. 

98. If 12 pears are worth 21 apples, and 3 apples cost 

a cent, what will be the price of four score and four pears? 


Ans, 49 cts. 
99. Ifa field will feed 6 cows 91 days, how long will it 
feed 21 cows ? Ans. 26 days. 


100. If 50 gallons of water in 1 hour fall into a cistern 
containing 230 gallons, and by a pipe inthe cistern 35 
gallons run out in an hour, in what time will it be filled ? 

, Ans. 15} hours. 

101. If 1 pint of wine cost 10 pence, how many & will 3 
hogsheads cost at that rate 4 Ans. 63£. 

102. If 1 gill of cider cost 3 pence, how many —& will 20 
gallons cost ? Ans. 8&. 

103. If 1 quart of vinegar cost 8 pence, how many —& will 
5 hogsheads cost ? Ans. 42£. 

104. If 4 pounds of nails cost 18 pence, how many & will 
12 tons cost ? Ans. 5044, 

105. If 12 tous of nails cost 504€, how many pence will 
4 pounds cost! Ans. 18 pence. 

106. If 3 hogsheads of wine cost 63, what will 1 pint 
cost ? Ans. 10 pence. 

107. If 4 pounds of iron cost 18 pence, how many dol- 
lars (New England currency) will 12 tons cost at the same 
rate ? : Ans. $1680. 

108. If 12 tons of iron cost 1680 dollars, (New England 
currency,) how many pence will four pounds cost ? 

Ans. 18 pence. 

109. If 18 pence will buy 4 pounds, how many tons can 
be purchased for 1680 dollars (New England currency ? 

Ans. 12 tons. 

110. If 1680 dollars'(New England currency) will buy 12 
tons of iron, how many pounds can be purchased for 18 
pence? Ans 4 pounds. 


SIMPLE PROPORTION. 89 


111. If 9 pounds of nails cost 6 shillings, (New York cur- 
rency, ) how many dollars will 30 tons cost ? 
| Ans. $5600. 
112. If 8 gallons of N. E. Rum cost 4 dollars, how 
many £ (N. Y.currency) will 15 pipes cost ? 
Ans. £378. 
113. If 4 qts. of oats cost 16 pence, (New Jersey cur- 
rency,) how many dollars will 60 bushels cost ? 
Ans. $8 53 33-L. 
114. If 3 pecks of beans cost 7 pence, (South Carolina 
currency,) how many dollars will 9 bushels cost ? 


Ans. $150. 

(115. If of a yard of cloth cost 6 dollars, what will 9 
yards cost ! 2 Ans, 72 dolls. 
116. If 2 of a yard of cloth cost 14 dollars, what will 4 
yards Gost? Ans. 64 dolls, 
117. If } of 2 ofa yard of silk cost 6 shillings, what will 

3 yards cost 1 Ans. 54s. 
118. If 4 of 3 oak yard of ribbon cost 18 cents, what 
will:7 yards énst 2 Ans. $2 52. 
119. If + of 2 of 2 of a yard of silk cost 50 cents, what 
will 4 of 8 ‘yards cost 4 Ans. 20 dolls. 
120. If 4 of 3 of 2 of a yard of satin cost 1 dollar, what 
will 2 of 4 of 5 yards cost ? Ans. 12 dolls. 
121. If 2 of 3 of 24 yards of cassimere cost 4 of 5 dollars, 
what will 2 yards cost? Ans. 4 dolls. 
122. If $o0f 2 of fof 6 vards of satinet cost 84 cents, 
what will ¢ of $ of 3 of 9 yards cost ? Ans. 892 cents. 


123. If 4 of + of 8 of 5%, of 34 yan of broadcloth cost 
4 of # of 4 of 4 dollars; what will ys of $ of 44 yards ney 
Ans: $ 1 05,0r 35. 
ae If 3 of 3 of #8, of 4 of 12 yards a petersham cost 

+ of 2 of of 10 dollars, what will 3 2 of 3) yards cost ? 
Ans. 94 dolls. 
125. If + of 7 yards of cloth cost 49 cents, what will £ 6 
of $ of 7 of 8 yards cost? Ans. $2612. 


90 COMPOUND PROPORTION. 


COMPOUND PROPORTION. 


Rule. In compound proportion, we have five terms 
given to find a sixth, three of which are a supposition, and 
two a demand. Plave the two terms of demand on the 
right hand side of the line, then place those in the suppo- 
sition of the same name directly opposite on the left, ob- 
serving to let the term of the same name of the answer 
stand on the right hand side of the line, and the last men- 
tioned. 


Note. When the effect of the cause is required, the 
question is in direct proportion, but when the cause is 
required, the question is in inverse proportion. Attention 
and practice will enable the pupil to apply the criterion 
with accuracy and facility. When the question ‘is inverse, 
all positions of the cause change place over the line. 
Observe, the effect never changes place, neither the an- 
swer required. Causes are men, horses, time, days, years, 
hours, capital or sum, length breadth, heighth, or any 
thing that produces an effect, as length, “brebdeh and 
heigth, are causes of solid contense. The effect is that 
pene is produced by the cause, as the work done, the 
grain consumed, the distance travelled, the money for the 
work, &c. By the positions above mentioned, we mean 
all the causes mentioned in the example, with the excep- 
tion of the one required. 


EXAMPLES. 


1. If 10 bushelsof oats be sufficient for 18 horses 20 days, 
how many bushels will serve 60 horses 36 days ? 


---18 | 60 

---20 | 36--- horses ~ 
| 10--- 
60 Ans. 


Note. In the above example itis required to know how 
many bushels of oats will serve the horses, consequently 
the effect of the cause is required; therefore the question 
is in direct proportion, and no positions change place. 


= 


SL eee 


COMPOUND PROPORTION. 91 


But had it been required to know how many horses 
would have consumed the oats in a given time, then the 
question would have been in inverse proportion, and the 
time would have been the position of the cause. 


2. If7 men can reap 84 acres of ryein 12 days, how many 
men will reap 100 acres in 5 days? 


As in direct: 


84 | 100 ---84 | 100--- 20 
12] 5 ---5 | 12--- positions 
ees | 7--- changed, viz. days. 


ae 


}: 20 meén, Ans: 


Note. In the above example we have two causes, viz., 
men and days, the men being required ; the other cause, 
viz., the days become positions, therefore change places. 

If the pupil will pay strict attention to the statement 
and solution of the six following examples, he will be 
enabled to determine at sight whether a question be in 
direct or inverse proportion. 

You may write the cause and effect under letters repre- 
senting them, the extremes being placed on the same 
side of the line, and the means on the opposite. Thus, 
take the first example : 


E C iE 

155-8 1088 beone<0 
20 36 
---18-4 608. 
&..20 ar 36--- 
Hoe 


| 60 Bushels. 


In the above it will be observed, that. in all examples 
under this rule, the supposition is full and complete, and 
the deficiency in the demand must be supplied with the 
answer when obtained: if your blank in the demand fall 
under the effect, then place your extremes fora divisor, 
and your means for a dividend; but ifthe blank fall under 
the cause, then place your means for a divisor, and your 
extremes for a dividend, as in the next example. 


V2 _COMPOUND PROPORTION. 


C E C E 
‘ 84 5 100 


Tp 0 
---84 | 100--- 20 
o-=e) “J 122-- 
> pe 
| 20 men. 


3. If 4 compositors, in 16 days of 12 hours long, can com- 
pose a work of 14 sheets, of 24 pages in each sheet, 44 
lines in a page, and 40 letters in a line; in how many 
days of 8 hours long may 12 compositors compose a 
volume, to be printed on the same letter, consisting of 42 
sheets, 16 pages on a sheet, 48 lines in a page, and 56 let- 
ters in a line ? 


C E C E 
4 14 8 42 
16 24° 12 16 
12 44 0 48 


40 55 
deel Aid Bou 
~a 2d |'16—- 4 
id, 4k | 48a 2 
40 | 55~+ 5-- 
ctl 2s hal Ya 
8 | 19 
168s 2 


| 24 Days, Ans. 


4. If 4 men can mow 12 acres of grass in 3 days, how 
many acres can 16. men mow in 9 days? 


wid, 16 
mini 
eee 


| 144 acres, Ans. 


ect i i i ta i, 


COMPOUND PROPORTION, 93 
5. If 4 men can mow 12 acres of grass in 3 days, how 
many men must be employed to mow 144 acres in 9 days ? 


—12] 14416 
aa f 
| 


Pe es 


—— 


| 16 men, Ans. 


6. If 16 men can mow 144-acres of grass in 9 days, how 
many men must be employed to mow 12 acres in three 
days ? 


—144 | 12— 4 
oe ee oa 


{16— 


ns 


| 4 men, Ans. 


7. If 16 men can mow 144 acres of grass in 9 days, inhow 
many days can 4 men mow 12 acres 2 ; 


—4| 16— 
—144| 12— 3 
pad 


| 3 days, Ans. 


8. If 16 men can mow 144 acres of grass in 9 days, how 
many acres can 4men mow in 3 days? 


—16} 4 
—-9} 3 
Riad 
|.,.d2 acres. 


9. If 4 men can mow 12 acres of grass in 3 days, inhow 
many days can 16 men mow 144 acres ? 


ac 9 i Be 
—12| 144— 9 
oe 


9 days, Ans. 


94 COMPOUND PROPORTION. 


10. If 6 men can build a wall 80 feet long, 6 feet wide, and 
4 feet high, in 15 days, what will be the length of that 
wall which 18 men can build in 30 days, the width being 
8 feet, and heighth 6 feet ? 


Statement. Solution. 
6 | 18 —6 | 18— 3 
15 | 30 —15 | 30— 2 
6; 8 —Ss } 6— 
4)6 —6 | 4 
| 80 | step 


240 Length, Ans. 


Note. Length being required, change width and 
heighth; width required, change length and heighth ; 
heighth required, change length and width. 

If 6 men can build a wall 80 feet long, 6 feet wide, and 
4 feet high, in 15 days, what will be the width of that 
wall which 18 men can build in 30 days, the length being 
240 feet, and the heighth 6 feet 4 


6 | 18 6 fF leas 
15 | 30 moat | 30— 2 
80 } 240 mie, 940 1 80-— 

4| 6 gi | A 

é be 


8 feet wide, Ans. 


11. If 6 men can build a wall 80 feet long, 6 feet wide, and 
A feet high, in 15 days, what will be the heighth of that 
wall which 18 men can build in 30 days, the length being 
240 feet, and the width 8 feet ? 


6/18 —6 | 18— 3— 
15 | 30 “a5, 30— 2— 
80 ; 240 —240 ; 80— 

6; 8 —s8 | 6 

i 


6 feet high, Ans. 


COMPOUND PROPORTION. 95 

12. If acellar 224 feet long, 17,3, feet wide, and 10} feet 

deep, be dug in 23 days by. 6 men, working 1255 four 

a day, how many days, of 85 hours, should 9 men ‘take to 

dig another, measuring 45 ee long, 342 wide, and 12,5 
feet deep ? 


Statement. Solution. A 

5} 41 —10!123— 3— 
23 {0 —41 |} 5— 

6|9 —9 16 

45 45— 

a4 2 —45 | 2— 

5 7 £73 —173 | 10— 
173° {310 —5 ; 173— 

161° $25 —41 |} 4—°2 

41-V 4 —2 —10 |] 123— 3— 

21 9d —2 | 6— 


| 12 days, Ans. 


13. If 8 horses consume 36 bushels of oats in 9 days, 
how many bushels will 24 horses consume in 12 days ? 
Ans, 144. 
14, If 8 horses consume 36 bushels of oats in 9 days, 
how many horses will be required to consume 144 bushels 
in 12 days ? Ans. 24 horses. 
15. If 8 horses consume 36 bushels of oatsin 9 days, in 
what time will 24 horses consume 144 bushels ? 
Ans. 12 days. 
16. If 24 horses consume 144 bushels of oats in 12 days, 
how many bushels of oats will 8 horses consume in nine 
days? Ans. 36 bush. 
17. If 24 horses consume 144 bushels of oats in 12 days, 
in what time will 8 horses consume 36 bushels? 
Ans. 9 days. 
18. If 24 horses consume 144 bushels of oats in 12 days, 
how many horses will be required to consume 36 bushels 
in nine days ? Ans. 8 horses. 
19. If6 men can build a wall 80 feet long, 6 feet wide, 
and 4 feet high, in 15 days, how many men must be em- 
ployed to build one 240 feet long, 8 feet wide, and 6 feet 
high, in 30 days? Ans. 18 men. 


96 COMPOUND PROPORTION. 


20. If 6 men can build a wall 80 feet long, 6 feet wide, 
and 4 feet high, in 15 days, in what time. can 18 men 
build one 240 “feet long, 8 feet wide, and 6 feet high ? 

Ans. 30 days. 

21. If 6 men can build a wall 80 feet long, 6 feet wide, 
and 4 feet high, in 15 days, what will be the length of 
that wall which 18 men can build in 30 days, the. pee 
being 8 feet, and height 6 feet ? Ans, 240 feet. 

22. If 6 men can build a wall 80 feet long, 6 feet wide, 
and 4 feet high, in 15 days, what will be the width of that 
wall which 18 men can buildin 30 days, the length being . 
240 feet, and the height being 6 feet ? ang: 8 feet. 

23. If6 men can build a aval! 80 feet long, 6 feet wide, 
and 4 feet high, in 15 days, what will be the height of that 
wall which 18 men can build in 30 days, the length being 
240 feet, and the width 8 feet ? Ans, 6 feet. 

24. If 18 men can build a wall 240 feet long, 8 feet 
wide, and 6 feet high, in 30 days, how many men must 
be employed to build one 80 feet long, 6 feet wide, and 4 
feet high, in 15 days 4 Ans. 6 men. 

25. if 18 men ean build a wall 240 feet long, 8 feet 
wide, and 6 feet high, in 30 days, in what time will 6 men 
build one 86 feet long, 6 feet wide, and 4 feet high ? 

Ans. 15 days. 

26. If 18 men can build a wall 240 feet long, 8 feet 
wide, and 6 feet high, in 30 days, what will be the length 
of that wall which 6 men can build in 15 days, the width 
being 6 feet, and height 4 feet? Ans. 80 feet. 

27. If 18 men can build a wall 240 feet long, 8 feet 
wide, and six feet high, in 30 days, what will the width 
of that wall be which 6 men can build in 15 days, the 
length being 80 feet, and the height 4 feet? Ans. 6 feet. 

28. If 18. men can build a wall 240 feet long, 8 feet 
wide, and 6 feet high, in 30 days, what will the height of 
that wall be which 6 men can build in 15 days, the length 
being 80 feet, and width 6 feet? Ans, 4 Rot, 

29. Lent a friend $800 for 6 months, and at the expira- 
tion of the time received the interest, which was 48 dol- 
lars, at whatrate per cent. per annum did I receive inter- 
est ? Ans. 12 cents. 

30, If 960 dollars defray the expenses of 20 men 88 


COMPOUND PROPORTION. O7 


weeks, for how. many weeks will $1440 defray the ex- 
penses of 48 men, if they spend at the same rate ? 
Ans. 55 weeks. 


31. Suppose 4 men in 12 days mow 48 acres, how 


many acres can 8 men mow in 16 days? 
Ans. 128 acres. 


32. If 10 bushels of oats be sufficient for 18 horses 20 
days, how many bushels will serve 60 horses 36 days ? 
Ans. 60 bushels. 


33. If 4 dollars be the hire of 8 men for 3 days, how 


many days must 20 men work for 40 dollars? 
Ans. 12 days. 


34. If 8 men can build a wall 24 feet lone, 14 feet 
high, and 4 wide, in 18 days, in how many days will 15 
men build a wall 175 feet long, 8 feet high, and 6 wide? 

| Ans. 60 days. 


35. If a footman travel 240 miles in 12 days, when the 
days are 12 hours long, how many days will be required 
to travel 720 miles, when the days are 16 hours long? 

Ans. 27 days. 

36: If 14 mencan dig a ditch 36 feet long, 7 wide, and 
8 deep, in 16 days, how many men will it take to dig 
another ditch 240 feet long,, 9 wide, and 5 deep, in 10 
days ? Ans. 120 men. 


37, If to make 24 yards of cloth 6 quarters wide it re- 
quires 12 ounces of wool, how much wool will it take to 
make 140 yards, 4 quarters wide ? Ans. 28 lbs. 


38. If300 men, in six months, perform a piece of work, 
when the days are 12 hours long, how many men will do 
the same in 4 months, when the days are 8 hours long ? 

wrk Ans. 675, 

39. If the transportation of 12 cwt. 3 qrs. for 400 miles 
cost 57 dollars 12 cents, what will the transportation of 
10 tons for 75 miles amount to ? Ans. $168,00 cts. 

40. An usurer put out 150 dollars at interest; and 
when it had been on interest 8 months, he received for 
principal and interest 160 dollars; at what rate per cent, 
per annum did he receive interest ? Ans. 10-per cent. 

41. If 8 men can build a wall 20 feet long, 6 feet high, 

9 


98 COMPOUND PROPORTION. 
we 


and 4 feet thick, in 12 days, in what time will 24 men 
build une 200 feet long, 8 feet high, 6 feet thick ? 
Ans. 80 days. 
42, Suppose 12 men consume 240 pounds of bread in 
8 days, how many men will consume 360. pounds in one 
day? . Ans. 144. 
43. How many men can complete a trench of 135 yards 
long in 8 days, when 16 men can dig 54 yards in 6 days % 
Ans. 30. 
44, If 16 bushels of oats serve 9 horses 6 days, how 
many bushels would 27 horses consume in 11 days ? 
Ans. 88. 
45. If a footman travels from New York to Boston, 
-which is~250 miles, in 8 days, when the days are 12 
hours Jong, in how many days. may he travel from New 
York to Charleston, South Carolina, which is 925 miles, 
when the days are 16 hours long ? Ans, 221, 
46. If 2 horses consume as much corn as 5 oxen, and 
12 horses consume 56 bushels in 20 days, how many bush- 
els will 18 oxen consume in 25 days ? Ans. 42. 
47. If 2 barrels of beer be sufficient to last a family of 
14 persons 24 days, how many barrels will be drunk out 
by a family of 24 persons in one year? Ans. 524. 
48. If 248 men, in 5 days, of 11 hours each, can dig a 
‘trench 230 yards long, 3 wide, and 2 deep, in how many 
days, of 9 hours each, will 24 men dig a trench 420 yards 
long, 5 wide, and 3 deep ? Ans, 2883%3,, 
_49. If 56 pounds of bread be sufficient for '7 men 14 
days, how much bread will serve 21 men 3 days ? 
Ans. 36 Ibs. 
50. If 4 reapers receive $11,04 cts. for 3 days’ work, 
how many men may be hired 16 days for $103,04 cts. ? 
- : Ans. 7. 
51. If 20 bushels of wheat are sufficient for a family of 
8 persons 5 months, how much will be sufficient for 4 
persons 12 months? Ans. 24. 
52. If 30 men perform a piece of work in 20 days, how 
many men will accomplish another piece of work 4 times 


as large, ina fifth part of the time ? Ans. 600. 
53. If '7 men can build 36 rods of wall in 3 days, how 


many rods can 2@ men buildin 14 days? Ans. 480. 


COMPOUND PROPORTION. . §9 


54. If40 men, in 10 days, can reap 200 acres of grain, 
how many acres can 14 men reapin 24 days? Ans. 168. 
55, If 4 men mow 96 acres of grass in 12 days, how 
many acres can 8 men mow in 16 days ? Ans. 256. 
56. Ifa family of 8 persons, in 24 months, spend 480 
dollars, how much would 16 persons spend in 8 months ? 
Aus. $320. 

57. If 7 quarts of malt are sufficient for a family of 7 
persons for 4 months, how many quarts are enough for 


46 persons 10 months? Ans. Td, 
58. If 8 reapers have £31 for 4 days’ work, how much 
will 48 men have for 16 days’ work ? Ans. 764. 


59. Ifa footman travels 240 miles in 12 days, when the 
days are 12 hours long, how many days may he travel 
720 miles in, of 16 hours long ? ATiee ac 

60. If 9 students spend in 18 days £102, how many 
dollars, New Jersey currency, will 63 students spend in 
30 days? ; Ans. 320. 

61. If 30 shillings be the hire of 8 men for3 days, 
how many days must 20 men work for £151 eins: £2. 

62. If 4 reapers have 24 shillings for 3 days’ work, how 


many men will earn £44 in 16 days? Ans. 3. 
63. If 9 men reap 18 acres in 3 days, how many acres 
will 27 men reap in 6 days 4 Ans. 108. 


64. If 25 men, by working 10 hours a day, can dig a 
trench 36 feet long, 12 feet broad, and 6 feet deep, in 9 
days, how many hours a day must 15 men work, in order 
to dig a trench 48 feet long, 8 feet broad, and 5 feet deep 
in 12 days? . . Ans. 95%. 

65. Ifa man travels 60 miles in 5 days, by travelling 3 
hours each day, how far will he travel in 10 days, by tra- 
velling 9 hours each day ? Ans. 360. 

66. If 5 men can build 10 rods of wall in 6 days, how 
many rods can 20 men build in 18 days ? Ans. 120, 

67. If 4 men receive 24 dollars for 6 days’ work, how 
much will 8 men receive for 12 days’ work ? Ans. $96. 

68. If 4 men receive 24 dollars for 6 days’ work, how 
‘many men may be hired 12 days for 96 dollars? Ans. 8. 

69. If 8 men, in 12 days, receive 96 dollars, how much 
will 4 men receive for 6 days’ work? = Ans. $24. 

70. If 8 men receive 96 dollars for 12 days’ work, how 
. long may 4 men be hired for 24 dollars? _* Ans. 6. 


100 COMPOUND PROPORTION. 


71. If 9 persons in a family spend 1512 dollars in one 
year, how much will 3 of the same family spend in four 
inonths ? Ans. $168. 

72. If 2000 dollars will support a garrison of 150 men 3 
months, how long will 6000 dollars support 4 times as 
many men ? Ans. 24. 

_ 73. If 144 men, in 6 days, of 12 hours each, dig a 
trench 200 feet long, 3 feet wide, and 2 feet deep, how 
many hours long is the day when 30 men dig atrench 350 
feet long, 6 feet wide, and 3 feet deep in 2593 days ? 

. Ans. 7. 

74. There was a certain edifice completed in one year, 
by 20 workmen; but the same being demolished, it is 
necessary that just such an one should be built in 5 
months; I demand the number of men to be employed 


about it ? Ans, 48. 
75. If 8 men spend 32 pounds in 13 weeks, what will 
24 men spend in 52 weeks ? Ans. £384. 


76. A wall, to be raised to the height of 27 feet, was 
raised to the height of 9 feet by 12 men in 6 days; how 
many men must be employed to finish it in 4 days ? 

ets Ans, 36. 

77. If6 laborers dig a ditch 34 yards long in 10 days, 

how many yards will 20 laborers dig in 15 days 4 
Ans, 170. 

78, Ifa garrison of 600 men have provisions for five 
weeks, allowing each man 12 oz. per day, how many men 
may be maintained 10 weeks, with the same provisions, 
if each manis limited to 8 oz. per day 4 Ans. 450. 

79, If 3 bushels and 3 pecks of corn will last a family of 
9 persons 22 days, in how many days will 6 persons con- 
sume 5 bushels? ' Ans. 44, 
_ 80. If 450 tiles, each 12 inches square, will pave my - 
cellar, how many tiles must I have if they are only 9 inches 
long and 8 inches broad? Ans. 900. 

81. If 12 ounces of wool make 24 yards of cloth, 6 
quarters wide, how much wool is required for 150 yards 4 
quarters wide ? Ans. 30 lbs, 
. 82. Ifabar of iron, 4 feet long, 3 inches broad, and 11 
inches thick weighs 36 lbs., what will a bar weigh that is 
6 feet long, 4inches broad, and 2 inches thick? Ans. 96. 


COMPOUND PROPORTION. 101 


83, If 14 men can reap 84 acres in 6 days, how many 
men will reap 44 acres in 4.days? 3 Ans. 11. 
84. If acistern 173 feet in length, 10% in breadth, and 
13 feet deep, holds 546 barrels of water, how many bar- 
rels will fill a cistern that is 16 feet long, 7 feet broad, and 
15 feet deep? Ans 384, 
85. If 25 pears can be bought for 10 lemons, and 28 
lemons for 18 pomegranates, and 1 pomegranate for 48 
almonds, and 50 almonds for 70 chestnuts, and 108 chest- 
nuts for 24 cents, how many pears can I buy for $1,35 cts.? 
; Ans, 3374. 
86. If the interest on 347 dollars for 34 years be 72 
dollars 87 cents, what will be the interest, at the same 
rate, on 537 dollars for 24 years? Ans. $80,55. 
87. What must be paid for the carriage of 4 cwt. 32 
miles, if the carriage of 8 cwt. 128 miles cost 12 dollars’ 
80 cents ? Ans, $160. 
88. By working 9 hours a day, 5 men hoed 18 acres of 
corn in 4 days, how many acres will 9 men hoe at that 
rate in 3 days, working 10 hours a day ? Ans, 27. 
89. One pound of thread makes 2 yards of linen cloth, 
5 quarters wide; how many pounds of thread will be re- 
quired to make 50 yards, 3 quarters wide ? Ans. 15. 
90. If 6 men, working 7 hours a day, mowed 28 acres 
of grass in 4 days, how many men at that rate will mow 
i6 acres in 8 days, working 6 hours a day ? Ans. 2. 
91. If5 men can make 300 pairs of boots in 40 days, 
how many men must be employed to make 900 pairs in 
60 days? - Ans. 10. 
92. If 3: compositors set 154 pages in 23 days, how 
many will be required to set 69? pages in 64 days? 
. Ans. 6. 
93. If the wages of 6 men for 14 days be 84 dollars, 
what will be the wages of 9 menfor 11 days? Ans. $99. 
94. If 3 lbs. of yarn make 9 yards of cloth, 5 quarters 
wide, how many lbs. will be required to make a piece of 
cloth 45 yds. long, and 4 quarters wide ? Ans, 12 Ibs. 
95. Ifa class of 25 boys perform 1750 examples in 
arithmetic in 15 hours, how many examples of equal length 
may a class of 30 boys perform in18 hours? Ans. 2520, 


102 COMPOUND PROPORTION. 


96. If the use of 100 dollars for 90 days be worth 1 dol- 
lar 50 cents, what is the use of 78 dollars worth for 85 
days ? Ans. 1104. 

97. If a man travels 217 milesin 7 days, travelling 6 
hours a day, how many miles will he travel in 9 days, if 
he travels 11 hours a day ? Ans. 5114. 

98. Ifa man performs.a journey of 1250 miles in 15 
days, by travelling 14 hours a day, how many days will it 
take him to perform a joumEy of 1000 miles by travelling 


13 hours a day ? Ans. 1212, 
99. If 10 cows eat 7} tons of hay in 14 weeks, how 
many cows will eat 224 tons in 28 weeks ? Ans. 15, 


100. If6 men will mow 35 acres of grass in 7 days, by 
working 10 hours a day, how many men will be required 
tomow 48 acres in 5.days, when they work 12 hours a 
day ? Ans. 9. 

101. If 16 men can build 18 rods of wall in 12 days, 
how many men must be employed to build 72 rods of the 
same kind of wall in § days ? Ans. 96. 

102. If 154 bushels of oats will serve 14 horses for 14 - 
days, how long will 406 bushels serve 7 horses ? 

Ans. 735% 

103. If 25 men can earn 6250 dollars in 2 years, how 
long will it take 5 men to earn $11250? Ans. 18 yrs. 

104, If9 men can mow 36 acres of grass in 4 days, how 
many acres willl9 men mowin 11 days? Ans, 209% 

105. If a family of 9 persons spend 450 dollars in 5 
months, how much would be sufficient to maintain the 
family 8 months, if5 more persons were added ? 

Ans. $1120. 

106. If astream of water, running into a pond of 190 
acres, will raise the pond 10 inches in 12 hours, how much 
would a pond of 50 acres be raised by the same stream in 
10 hours ? _Anse312. 

107. If 725 bottles hold 4 barrels of wine, how many 
bottles are required to hold 3 tierces of wine ? 

Ans. 725, 

108. If 12 men can build a brick wall 25 feet long, 7 feet 
high, and 4 feet thick in 18 days, in how many days will 
20 men build a brick wall 150 feet long, 8 feet high, and ' 
5 feet thick ? Ans. 924. 


> 


COMPOUND PROPORTION. 103 


109. If 15 men can dig atrench 75 feet long, 8 feet wide, 
and 6 feet deep in 12 days, how many men must be .em- — 
ployed to dig a trench 300 feet long, 12 feet wide, and 9 
feet deep in 10 days ? | Ans. 162. 

110. If 175. bushels of corn, when corn is worth 60 “cts. 
per bushel, be given for the carriage of 100 barrels of 
flour 58 miles, how many bushels of corn, when corn is 
worth 75 cents per bushel, must be given for the carriage 


of 90 barrels of flour 200 miles ? Ans, 4344. 
111.. How many acres in a piece of Jand 560 rods long 
and 32 rods wide ? Ans. 112. 


112. How many yards of carpeting that is 2? of a yard 
wide, are sufficient to cover a floor that is 18 feet wide and 
60 feet long ? Ans. 160 yds. 

113, What is the weight of a pea to asteelyard, which, 
being suspended 39 inches from the centre of motion, will 
equipoise 208 lbs. suspended at the draught end ? of an 


inch ? Ans. 4 lbs. 
114, If 17 tons 12 cwt, of iron cost $880, what cost 2 
cwt.? Ans. $5. 


115. A borrowed of B $250 for 7 months; and, in re- 
turn, lent him $300; how long ought he to keep it, that 
the interest of it may be equal to that of the first sum? 

Ans. 52 mo. 
- 116. If 2of a-yard of cloth, 4 yard wide, cost £2, what 
is the value of 3 of a yard, 1# yards wide, of the same 
quality ? Ans. 131 shillings. 

117. If £600 principal gain £334 interest in 10% 
months, in what time will £100 gain £642? Ans. 12 mo. 

118. If2 men in of a year expend $564, how much 
will defray the expenses of 3 persuns for 54 years at the 


same rate? ; Ans. $600. 
119. How many men can do as much work in ;4 of a 
month as 16 mencan doin 14 months ? Ans. 60. 


120, What sum has A at interest, when it yields as 
much in 74 months as B’s $450 does in 15 months? 


; Ans. $900, 
~121. When 12 oxen graze 164 acres of grass in 20 days, 
how much will suffice 24 oxen 100 days? Ans. 1624, 


122. What is the freight of 10,000 bricks from Waldo- 
boro’ to Boston, at $1,25 cts. per 2000 lbs., allowing 6 
bricks to weigh 264 lbs. ? Ans. $27,60 cts. 


* 


104 SIMPLE 1NTEREST. 


123. If a man receive $15 for 1 year’s interest of money 
lent,.at 6 per cent. per annum, how much was the sum 
lent ? Ans. $250. 

124, If 8 boarders drink a barrel of cider in 12 days, 


‘how long would it last, if 4 more came among them? 


Ans. 8 days. 
125. When wheat is sold at 93 cents per bushel, the 
penny loaf weighs 12 ounces; what must it weigh when 
the wheat is $1,24 cts. per bushel? Ans. 9 ounces. 
126. How many yards of baize, ? wide, will line a cloak 
which has in it 12 yards of camlet, 3 a yard wide? 
Ans. 8 yds. 
127. Suppose 400 men in a garrison are provided with 
provisions for 30 days; how many men must be sent out, 
if they would have the provisions last 50 days? Ans. 160. 
128. If a head of 7 feet of water with 30 mill powers 
will reduce a pond of 200 acres 8 inches in a day, how 
much will ahead of 6 feet reduce it in the same time? 
| ; Ans. 94, 
129. If 30 mill powers would reduce a pond of 7 feethead 
7% inches in a day, how much would they reduce a pond 
of 6 feet 62 inches head inthe sametime? Ans. 8 inches. 
130. A ship’s company, of 15 persons, is supposed to 
have bread to last their voyage, allowing each 8 ounces 
per day: when they picked up a crew of 5 persons in dis- 
tress, to whom.they are willing to communicate, what will 
the daily allowance of each person then be ? 
Ans. 6 ounces, 
131. A person engaged to remove 800 tons of timber 
from Exeter to the navy yard, in Portsmouth. If in 6 
days he has removed 450 tons with 36 oxen, how many 
oxen would be wanted to remove the remainder in 3 
days? Ans. 56 oxen. 


SIMPLE INTEREST. 


Rue. Place the principal, time and rate per cent. on 
the right hand side of the line. If the time consists of 
years and months, reduce them to months, and place 12, 
(the number of months in a year) on the left hand side of 


SIMPLE INTEREST. 105. 


the line. Should the time consist of months and days, re- 
duce them to days, or aliquoit parts of a month. If reduced 
to days, place 30 (the number of days in a month) and 
12 on the left. Ifto aliquoit parts of a month, place 12 
only, as above. 


EXAMPLES. 


1. What is the interest on $60, for 117 days, at 6 per 
cent? 


60— 2— 
---30 | 117 
mceie't Be. 
| $1,17 Ans. 


Norr. When the principal is in dollars and cents cut 
off 4 figures in your answer from the right for cents, and 
all to the left are $s; but if your principal be $s only, cut 
off 2 figures. 


2. What is the interest of $96,50 for 90 days, at 6 per 
cent? 
96— 50— 4825 
90— 3 
Gi 


—30 
---2 ---12 


| $1,44,75 Ans. 


3. What is the interest of $4,80 for 361 days, at 6 per 
cent. ? 


4,80— 4 
—30 | 361 
—12 | 6— 2 


| $,28,88 Ans. 


What is the interest of $720 for 9 months, at 7 per 
cent.} : 


720— 60 
—12|9 
7 


| $37,80 Ans. 
10* | 


106 SIMPLE INTEREST. 


5, What is the interest of $960 for 11 months and 20 
days, at 6 per cent. ? ‘ 


960— 80 
aii 359 
athe {eS aaa 

| $56 Ans. 


Nore. In the above example we say 11 months and 
20 days make 113 months, 20 days being % of 30, this 
mixed number we reduce to an improper fraction, making 
35months, writing the numerator on the right, and the 
denominator on the left of the line. 


6. What is the interest of $144,60 for 5 years, 11 
months and 27 days, at 6 per cent. ? 


rs. mo. ds. mos. 


eri 27=712, 144,60— 723 
Fs ——1034..719 
—- —2--12 | 6— 
10 sie 


| $51, 98,37 Ans. 


8. Required the interest of 180 dollars for two months 


and 15 days, at 6 per cent. Ans. $2 25. 
9, Required the interest of 180 dollars for 6 months, at 
6 percent... Ans. $5 40. 
10. Required the interest of 120 dollars for 8 months, at 
33 per cent. Ans. $2 66,6-+-. 
11. Required the interest of 99 dollars for 2 months, at 
44 per cent. Ans. $0 79,2. 
12. Required the interest of 60 dollars for 4 months, at 4 
per cent. Ans. $ 0 80. 
13. Required the interest of 176 dollars for 3 months and 
10 days, at 6 per cent. Ans. 2 93,3-++. 
14. Required the interest of 640 dollars for two months 
and 19 days.* Ans. $8 42,6, 


* When no per cent. is mentioned, 6 per cent. is always understood. 


SIMPLE INTEREST. 107 


15. Required the interest of 800 dollars for 4 months and 


20 days. Ans, $18 66,6-L. 
16. Required the interest of 720 dollars for 3 months and 
19 days. Ans. $13 08. 
17. Required the interest of 480 dollars for six months 
and 5 days. Ans. $ 14 80. 
18. Required the interest of $480 for 8 months and 6 
days. Ans, $ 0 19,68, 
19. Required the interest of $720 for 9 months and 9 
days. : Ans. $ 0 33,48. 
20. Required the interest of 480 dollars for 1 year 6 
months and 15 days, at 6 per cent. Ans $44 40. 
21. Required the interest of $19 20 for 2 years 4 
months and 10 days. Ans. $2 72. 
22. Required the-interest of $384,40 for 8 years 2 
months and 20 days. Ans. $74 31,73. 
‘23. Required the interest of $9999 for 2 years 6 months 
and 20 days. Ans. $15 33,18, 
24. Required the interest of $600,48 for 2 years 9 
months and 29. days. _.. Ans. $101 98,15-++. 
25. Required the interest of 480 dollars for 6 years 6 
months and 6 days. Ans. $187 68. 
36. Required the interest of $960 60 for 4 years 4 
months and 15 days, Ans. $252 15,75. 


When interest is required on notes or bonds on which 
partial payments have been made. 


Rule. Cast the interest on the principal at the given 
rate per cent., up tothe time of the first payment, then, 
if the payment exceed the interest, deduct the excess from 
the principal; but if it be less, set both payment and in- 
terest aside, and cast.the interest on the same principal to 
the time of the next payment, or to the time of some pay- 
ment, which, when added to the preceding payments, will 
exceed the sum of interest then due, and deduct the sum 
of these payments from the amount of the principal. The 
remainder will form anew principal, with which proceed 
as befure, till the time of settlement. 9 


108 SIMPLE INTEREST. 


EXAMPLE, . 


87. For value received, I promise to pay John Smith 
& Co., or order, fifteen limdred dollars on demand, with 
interest. | JOHN YORK. 

January Ist, 1825. 

On this note are the following endorsements : Oct. Ist, 
1825, three hundred dollars; July 1st, 1827, four hun- 
dred and fifty dollars ; Sept. 1st, 1828, six hundred and 
fifty dollars. 

What was due on settlement, J uly ist, 1830? 


YT. Whe = 
1825 10° 1 
1825" 4.1 


9 0 time till first payment. 
1500— 750 
—2 —12 
62 s 


SSS pS 


| $67,50 The interest till 
first payment, and $1500-+67,50—1567, 50 amount, = 
1567,50—3800=$1267,50 the new principal. 


ys cme. ds 
LB 29 Fe Vk : 
: 182510," oL 
1 9 O=time from Ist to 2d pay- 
ment. 
126750— 63375 
—2 —12 | 21 
ale 


$133,08,75 The interest 
till 2d payment, then 126'7,50+-13308,7,5=1400,5875, and 
1400,587 —450—950,587 the new principal. 

1825..9.-4 

1627 AFSL 


1 2 0 time from 2d to 3d pay- 
ment. ; 


SIMPLE INTEREST. 109 


| 950 587 
—2 —12 | 14— 7 


foot Bs | 66,54,109 The interest 
till the 3d payment, and 950,587-+-66,541—1017,128, and 
1017,128—650= 367,128 the new principal. 


1830. 7 1 
Leo Oe 
110 0 time from 3d payment to 
settlement. 
| 367,128 
ent AO oo if 
| 6— 


| 40,384. The interest till 
time of settlement and 367,128-+-40,384—407,512. Ans. 
or sum due on settlement. 


38. I have in my possession a note dated April 15, 1833, 
for $2150,25, on. which are the following endorsements : 

Noy. 8th, 1834, $500; Sept. Ist, 1835, $72364; Janua- 
ry Ist, 1837, $378,295, and Oct. 29th, 1837, $850. What 
amount was due April 15th, 1838? Ans. $138,337. 


39. On a note of $767,95, given Dec. 25, 1827, and draw- 
ing interest after 90 days, were the following endorse- 
ments: January Ist, 1830, $75; March 25, 1831, $565,25. 

What was due January Ist, 1833 ?. Ans. $294,118, 
* 40. What was due on anote of $2100, dated June 15, 
1820, on settlement June 15, 1830. The following sums 
being endorsed on the back of it: June 30, 1824, $750, and 
Sept. 30, 1828, $1200 on interest, at 6 per cent? 

“Ans, 1249,527. 


41. For value received of A.M., I promise to pay him or 
order seven hundred and fifty dollars, with interest, at 6 
per cent. 

January 1, 1824. G. G. G, 

On the above were the following payments, endorsed 
April 1, 1826, $150; July 1, 1829, $450, What was due 
on settlement Sept. Ist, 1832 2 Ans. $461,71, 


110 CONJOINED PROPORTION. 


CONJOINED PROPORTION. 


Route. When it is required to find how many of. the 
Jirst sort of coin, weight or measure, mentioned in the 
question, are equal to a given quantity of the last, place 
the numbers alternately, beginning on the right hand side 
of the line, observing to let the last number stand on the 
right hand side; but when itis required to find how many 
of the Jas¢ sort are equal to a given quantity of the jst, 
place the numbers alternately, beginning on the /eft hand 
side of the line, and let the last number stand on the right 
hand side. 

1. If 100 lbs. English make 95 lbs. Flemish, and 19 Ibs. 
Flemish make 25 lbs. at Bologna, how many lbs. English 


are equal to 50 lbs. at Bologna ? 
—5 —95 | 100— 4 


mee) 1 1 me 
hon) 
e | 40 lbs. Ans. 


2. If 40 lbs. at New York make 48 lbs. at Antwerp, and 
30 Ibs. at Antwerp make 36° lbs, at Leghorn, how many 
Ibs. at New York are equal to 144 lbs. at Leghorn? ~ 

—4 —48 | 4— 0 


+ 100 lbs. Ans. 


3. If 17 lbs. of raisins are worth 20 lbs. of almonds, 
and 5 lbs. of almonds worth 84 lbs. of figs, and 37% lbs. 
of figs worth 30 lbs. of tamarinds, how many lbs, of tama- 
rinds are equal in value to 424 lbs. of raisins ? 


~ 


—17 | 20— 
- 5 | 17— 
—2 
—5 —15 —75 | 2 
30— 6— 2 
—2 | 85— 17 
: 68 lbs, Ans. 


_ 4, If A can do as much work in 3 daysas B can doin 44 
days, and B as much in 9 days as C in 12 days, and C as 


CONJOINED PROPORTION. 111 


much in 10 days as D im 8 days, how many days work of | 
D are equal to 5 days work of A. 


wi sac 
12— 2— 
i 9 8 
SMe: jo 
srarl io v9 Ans. 


5. If 6 braces at Leghorn make 3 . ells English, 5 ells 
English make 9 braces at Venice, how many braces at 
Leghorn will be equal to 90 braces at Venice? Ans. 100. 

6. If 3 dozen pairs of gloves be equal in value to 2 
pieces of Holland, 3 pieces of Holland to 7 yards of satin, 
6 yards of satin to 2 pieces of Flanders lace, and 3 pieces 
Flanders lace to 81 shillings, how many dozen pairs of 
gloves may be bought for 28 shillings ? Ans. 2. 

7. If 20 braces at Leghorn be equal to 11 vares at Lis- 
bon, and 40 vares at Lisbon to 80 braces at Lucca, how 
many braces at Lucca areequal to 100 braces at Leghorn ? 

Ans. 110. 

8. Suppose.100 pounds of Venice weight is equal to 70 
pounds of Lyons, and 60 pounds of Lyons to 50 pounds 
of, Rouen, and 20 pounds of Rouen to 25 pounds of Tou- 
louse, va 50 pounds of Toulouse to 37 pounds of Geneva, 
how many pounds of Geneva are equal to 25 pounds of 
Venice ? Ans. 1342, 

9. If one French crown is equal in value to 80 pence of 
Holland, and 83 pence of Holland to 48 pence English, 
and 40 pence English to 70 pence of Hamburgh, and 64 
pence of Hamburgh to 1 florin of Frankfort, ; how many 
florins of Frankfort are equal to 166: French crowns ? 

Ans. 210. 

10. If 70 braces at Venice are equal to 75 braces at 
Leghorn, and 7 braces at Leghorn are equal to 4 yards of 
the United States, how many braces at Venice are equal 
to 64 yards in the United States | Ans. 104,%;. 

11. A merchant in St. Petersburgh owes 1000 ducats 
_ in Berlin, which he wishes to pay in rubles by the way of 
Holland; and he has for the data of his operation the fol- 
lowing information, viz., that one ruble gives 474 stivers ; 
that 20 stivers make one —_ 25 florins 1 rix dollar of 


112 INSURANCE, 


Holland, that 100 rix dollars of Holland fetch 142 rix dol- 
lars of Prussia, and that 1 ducat in Berlin is worth 3 rix 
dollars Prussian ; how many fubles will pay the debt? . 
Ans. 2223153. 
12. If 94 piasters at Leghorn are equal to 100 ducats at 
Venice, and 1 ducat is equal to 320 maravadies at Cadiz, 
and 272 maravadies are equal to 630 reas at Lisbon, and 
400 reas are equal to 50 pence at Amsterdam, and 56 
pence are equal to 3 francs at Paris, and 9 francs are 
equal to 94 pence at London, and 54 pence are equal to 
a dollar in the United States, how many dollars are equal 
to 800 piasters ? Ans. 816432. 
13. If 140 braces at Venice be equal to 150 braces at 
Leghorn, and 7 braces at Leghorn be equal to 4 American 
yards, how many American yards are equal 527’, Vene- 
tian braces ? “Ans. 32. 
14. A merchant in London has credit for 500 piasters 
in Leghorn, for which he can draw directly at 52 pence 
sterling per piaster: but choosing to have it remitted by 
a circular route, they are sent, by his order, to Venice at 
95 piasters for 100 ducats banco; from thence to Cadiz at 
350 maravadies per ducat banco; from thence to Lisbon 
at 630 reas per piaster of 272 maravadies; from thence to 
Amsterdam at 48 pence Flemish for 400 reas; from thence 
to Paris at 54 pence Flemish per crown ; and from thence 
to London at 30 pence sterling per crown. What is the 
arbitrated price between London and Leghorn per piaster, 
and what is gained or lost by this circular remittance, 
without reckoning expenses ? 


Gained by circular remittance, £10 3s. 84d. 
Arbitrated value of a piaster by ditto.  S564277.d4 


EE IIE IT EY ET I I LI ET TD ET PET TE TLS LETTE ET ET EL OE EE TEE EET 
. INSURANCE. 


Rule. Place the value of the property insured and the 
rate per cent. on the right hand side of the line, and 100 
on the /eft hand side. 


1. Required the insurance on an East India ship and 
cargo valued at 124000 dollars at 122 per cent. 
Solution. —2 —100 | 124, 000— 15,500 dolls. Ans. 


F COMMISSION. 113 


2. Required the insurance on 72000 dollars at 44 per 

cent. Ans. $3456. 

3. Required the insurance on the ship Constitution and 
cargo, valued at yeen0g dollars at 62 per cent. 


Ans. $9600. 
4, Required ae insurance on 4 buildings, each valued 
at 2800 dollars at 31 per cent. : $3732, 


5. Required the insurance on the ship Elizabeth Ann, 
and cargo; the value of the ship being 80,000, and 
that of the cargo 4 of 2 of 4 of 2 of 2 of 7 of $ of the value 
of the ship at 4 per cent? ? " Ans. $4320. 

& Required the insurance of the brig Hannah, and 
cargo, valued at $160,000 at 142 per cent ? 

Ans. $230 40. 


COMMISSION. 


Rule. Place the value of the property deposited, and 
the rate per cent. of the commission on the 77ght hand side 
of the line, and 100 on the /eft hand side. : 


1. Required the commission on 800 dollars at 4 per 


cent. 
Solution. —100 | 800— 8 
4 
$32 Ans. 


a: A gentleman received goods to the value of 1200 dol- 
lars to be sold on commission at 33 per cent. Required 


his commission. Ans. $40. 
3. Required what a factor may demand on 44 per cent. 
commission for laying out $848,50. Ans. $40 72,8. 
4, Required my commission on 34 per cent. for $150 40. 
Ans. $5 371. 
5. Received goods to the value of 9000 dollars at 24 per 
cent. Required my commission. | Ans. $225. 


6. A gentleman deposited in my care goeds to the 
value of 17400 dollars, and allowed me 83 per cent. com- 
mission, with which I purchased other goods to the value 
of 1450 dollars. How much had I left ? Ans, nothing. 


114 EQUATION OF PAYMENTS. “ 


DISCOUNT. 


Rule. Place the sum on which*the discount is to be 
made on the 7v2ght hand side of the line, and the amount 
of one dollar for the given time and rate per cent on the 
left hand side, and the quotient will be the present worth. 
Subtract the present worth from the sum due, and you 
will obtain the discount. 


1. What is the present worth of 600 dollars, due 4 
years hence, at 5 per cent? " 


Solution. 100 s— $20 | 600—-5 


5) 100 
500 $500 Ans. 
4 
2000 
100 
120 


2, What must be discounted for the ready payment of 
100 dollars, due a year hence, at 6 per cent a year? 
Ans. $5 66. 
3. Bought goods amounting to $615 75, at 7 morths 
pinched much ready money must I pay, discount at 
$ per cent. per annum ? us Ans. $600. 
a4. What is the difference between the interest and dis- 
count on $600 for 12 years, at 5 per cent 4 
Ans. interest $360, discount $225, diff. 8135. 
5. What is the present worth of 672, due 2 years 
hence, discounting at the rate of °6 per cent. per annum? 
Ans. £600. 


EQUATION OF PAYMENTS. 


Rule. Multiply each payment by the time which must 
elapse before it becomes due, and place the sum of the 
products on the right hand side of the line, and the sum 
of the payments on the /eft hand side. 


BARTER. + 115 


1. A owes B $380, to be paid as follows, viz :—$100 in 
6 months, $120 in 7 months, and $160 in 10 months; 


what is the equated time for the payntens of the whole 
debt? 


Solution. 100+ 6= 600 
120 (== -740 
160-- 101600 


—380 | 3040—8 Ans. 8 months. 


2. A merchant has owing him £300, to be paid as fol- 
lows—2£50 in 2 months, £100 in 5 months, and the rest 
in 8 months ; and it is agreed to make one payment of 
the whole ; I demand the equated time. Ans. 6 months. 

3. F owes H $1000, whereof $200 is to be paid down, 
$400 in 5 months, and the rest in 15 months, but they 
agree to make one payment of the whole; when must 
’ that time be? Ans. 8 months. 

4. A merchant has due to him acertain sum of money, 
to be paid one-sixth in 2 months, one-third in 3 months, 
and the rest in 6 months; what is the equated time for 
the payment of the whole? Ans. 44 months. 


BARTER. 


“Rule. Place the several constituents of the commodity 
whose value is given on the right hand side of the line, 
and the céveeiurents of those whose value is required on 
the left hand side. 


Nore. The principle involved in this rule is the same 
as that in the Rule of Three or Simple Proportion. 


EXAMPLES. 


1. A has 120 bushels of wheat, worth 80 cents per bushel, 
for which B gave him 60 bushels of corn, what was the 
corn per bushel? 

80 
—60 | 120— 2 


| $1,60 Ans, 


116 . - BARTER, 


2. G has 12 hhds. of molasses, each hhd. containing 120 
gallons, valued at 40 cents per gallon, for which S gave 
him cloth valued at $4.80 per yard, how many yards of 
cloth will pay for the molasses ? 

—4,80 | 122— 
120 
wa 


| 120 yards. Ans. 


3. How much wheat at $1,25 a bushel must be given 
in barter for 50 bushels of rye, at 70 cents per bushel ? 
50— 2 As 
—5 —125 | 70— 14 


| 28 bushels. Ans. 


4, How much butter at 123 cents a lb. must be given in 
exchange for 12 lbs. of indigo, at $2,25 per lb. ? 
12 
225— 9 
2 


a pa: 


| 216 lbs. Ans. 


g 


5. Ahas 4 tuns of wine worth 3 pence per pint, New 
York currency, he will barter with B. and take rye at 63 
cents per bushel, how many bushels of rye will pay for 
the wine ? 


| 400 bushels. Ans. 


BARTER, e 117 


6. Bhas 12 tons of iron at 4 pence per lb., (New York) 
for which C. gave him wheat at $1,40 per bushel, how 
many bushels of wheat did C give for the iron ? 

12— 
| 20— 
Aine 
| 28— 2 
—12|4 
100 
—140 | ~ 


; | 800 bushels. Ans. 


”, A has cloth that cost 28 cents, B has cloth that cost 
him 22 cents, and he puts it at 25 cents, how high must 
A put his to gain 10 per cent. more than B? 


ee eee ay 
5-2 199") a5) 5 
Soy oy a 10s 

| 35 ects. Ans, 


_§. Bhas C’s note for $250 with 6 months interest due 
on it, and to redeem it, C delivers him 60 bushels of wheat 
at $1,25 per bushel, 50 bushels of corn at 874 cents per 
bushel, and the balance in staves. at $30 per thousand, 
what number of staves did B receive ? 

Ans. 5550 staves. 

9. A has tea which he bartered with B at 10 pence 
per lb. more than it cost him, against cambric which 
stands B in 10 shillings per yard, but he puts it at 12s. 6d, 
I would know the first cost of the tea? 

Ans. 3 shill. 4 pence per lb. 

10. A has 240 bushels of rye, ie cost him 90 cents 
per bushel, this he barters with B at 95 cents for wheat 
that stands B in 99 cents per bushel, how many bushels 
of wheat is he to receive in barter, had at what price is it 
to be rated, that their gains may be equal? 

Ans. 218,35, bush. at $1,043 per buss 

11. A and B. barter some goods, A puts his at 30,9, 
shillings, and gains 8 percent., B puts his at 2475 shillings, 
and gains at the same rate. What was the first cost of 
the goods? Ans. 28 shillings and 22 shillings 6 pence. 


~ 


118 ba _ BARTER. 


12. How many bushels of wheat at 5 shillings per 
bushel must I give for 84 bushels of corn at 7 shillings 6 
pence per bushel? Ans. 126 bushels. 

13. A buys of B 4 puncheons of rum, containing 410 
gallons, at $1,10 per gallon, and in return he pays him 
$112 cash, and the remainder in wheat, at $1,84 per 
bushel, how many bushels must ‘A receive ? 

Ans. 184,239, 

14. How much tea, at 7 shillings 6 pence per lb. must 
be given in barter for 234 yards of flannel, at 3 shillings 9 
pence per yard ? Ans. 117 lbs. 

15. Sold goods tothe value of $245, and received in 
payment 101 bushels of corn at 64 cents per bushel, the 
remainder is to be paid in wheat, at $1,12 per bushel, how 
many bushels will pay the balance ? Ans. 16154. 

16. Sold 96 yards of cloth at 4 shillings 8 pence, New 
York currency, per yard, received in payment 72 gallons 
of whiskey at 3 shillings 9 pence, New England currency, 
per gallon, how many dollars will pay the balance ? 

Ans. $11. 

17. A holds a note against B for $250, with 6 months 
interest thereon at 6 per cent., and to redeem it B de- 
livers him 70 bushels of wheat at 7 shillings 6 pence, New 
York currency, per bushel, and 250 bushels of corn at 4 
shillings 9 pence per bushel, how many dollars will pay 
the balance ? | Ans. $43,4375. 

18. How many yards of carpeting at 75 cts per yard, 
cash price, must be given for 45 yards of broadcloth at 
$5,80 per yard, bartering price, when the cash price was 
only $4,90. Ans. 2821-+ yards. 

19, A has 200 yards of cloth at 25 cents per yard ready 
cash, which he barters with B at 31 cents, taking sugar of 
him at 10 cents per 1b., which is worth but 8 cents, who 
gets the best bargain ? 

Ans. B gets 40 cents the best bargain. 

20. A has linen cloth at 30 cents per vard, ready cash, 
but in barter will have 36 cents ; B has 1810 yards of rib- 
bon at 22 cents per yard, ready cash, and would have 
$200 in present cash, and the rest in linen cloth. What 
price does the ribbon bear in barter per yard, and how 
much linen must A give B? Ans. The bartering price for 
the ribbon is 26 cts. 4 ms., B must receive 660% yards. 


° fot 
PROFIT “AND LOSS. 119 


21, Suppose A has 350 yards of linen ‘at 25 cents per 
yard, which he will barter with B for sugar at $5 percwt., 
how much sugar will the linen come to? 

Ans. 173 cwt. 

22. A and B barter, A has 3 cwt. 2qrs. 14 lbs. of cheese 
at $6 per cwt., but in barter he will have $64, B has but- 
ter worth 12 cents per lb., how must B sell his butter per 
lb. in order that he may sustain no loss ? Ans. 13 cents. 

23. How many lbs. of butter at B’s bartering price will 
pay him for 24 cwt. of his cheese at the above rates ? 

| Ans, 1124. 

24. Twomerchants, A and B, barter, A has 30 cwt. of 
iron at $5,40 percwt., B has 12 pieces of broadcloth, each 
piece containing 16 yards at $1,05 per yard, how many 
yards of carpeting at $1,05 per yard will pay the balance ? 

/ Ans. 5862 yards, 

25. A and B barter, A has 7 cwt.3 qrs. 14 Ibs. of cheese 
at $6 per cwt., but in barter he will have $6,72, B has 
satinett at $2 per yard, cash price, how must B sell his 
satinett in barter per yard tu be equal to A’s bartering 
price, and how many yards of satinett will pay for B’s 
cheese? Ans. $2,24 B’s bartering price, and 23 yards 2 
qrs. 2 nails pays for his cheese. 


PROFIT AND LOSS. 


Rule. When it is required to know what is gained or 
lost per cent., ascertain what the gain or loss is by sub- 
traction, then place 100 and the gain or loss on the rieutT 
hand side of the line; and the price it cost on the LEFT 
hand side. When itis required to know how a commodity 
must be sold to gain or lose so much per cent., place the 

-value of the commodity and 100 with the gain per cent., 
added, or loss per cent., subtracted on the rieuT hand side 
of the line, and 100:on the terr. When there is gained 
or lost per cent., to ascertain what the commodity cost, 
place the price at which it is sold and 100 on the RicuT 
hand side of the line, and 100 with the gain per cent. 
added or loss per cent. subtracted, onthe LerT hand side. 
When any commodity is sold at a given rate, and by which 


» 


120 PROFIT AND LOSS. 


so much can be gained or lost per cent., to know what 
would be gained or lost per cent., ifsold at any other rate, 
place the First price on the Lerr hand side of the line, 
-and the other price with 100 and the profit per cent. added 
or loss per cent. subtracted on the rieut hand side of the 
line. 

EXAMPLES. 


1. A gentleman purchased wine at 80 cents per gallon, 
how must he sell it per gallon to gain 20 per cent? 


100 | 8,0— 
Gain 20 1 —00 | 12,0— 
120 | 96 cts. Ans. 


2. A merchant bought sugar at 10 cents per lb., but 
being damaged, he was content to lose 20 per cent.; how 
much must he sell it per Ib. ? 


100 aot 
Loss 20 1 —00 : 8,0— 
80 | 8cts. Ans. 


3. Bought coffee at 12 cents per Ib., and sold it at 16 
cents per |b., what is the gain per cent ? 


Note. We will solve this example by twomethods; in 
the first method, we subtract the cost price from the selling 
price, and then place the cost price on the left hand side, 
and the difference between the two prices, with 100 on the 
right hand side of the line. Inthe second method, we 
place the cost on the left, and the selling price with 100 
on the right, the excess or deficiency of the 100 is the 
gain or loss per cent. is , 


16, 3-12 7100. 3 12} 100 
‘Sebpetonas fe ae Gee 4k = 


PROFIT AND LOSS. 121 


4. If 9 yards of cloth cost 63 dollars, how must 45 
yards be sold, to gain 40 per cent. ? , 


eon 


Rage Sh be 
wn Lee LOO HERE 
0% 
| $441 Ans. 


5. If 42 yards of cloth cost $6;%, how must 3 of 5 of $ 
of 5% yards be sold to lose 12 per cent. ? 


ey 0 Cah Seas 
nae ee er ae 
4 8 6a 
= | 1i— 
24 Lae 
1 IE 3 
5 —20 100. 88—:22—, 11 


emer Here 


20, | 33=$148, Ans. 


6. If 4 of 3 of a yard of linen cost {5 of 3 of a £, New 
Jersey currency, for how many cents must lof # of a 
yard be sold to gain 25 per cent. ? 

—4|1 : 
a | 4 
—4/);5— 
SN ig ghee Ml 
—5 —10 | 7— 
HR Fe Me Pc se 
N.J, —3|8,00— §$ 
—100 | 125— 5 


= 


| 40 cts. Ans. 


Il 


122 PROFIT AND LOSS. 


7. If 4 of 3 of £ of 5 of a yard of — cost 7, of 3 
of 4 of 8 saith a & (New agai currency,) for how many 
cents must ys of £ of 2 of ;3, of a yard be sold to gain 30 
per cent.7? 


le 19s 
pe hos 
PES FS. I: A oe 
—10:| a 
pleas Ive Be 5 aed 
Pate 5 it? ‘al < Peasy 
6 | 7 
—7 fds 
: EO ape 
a Pisa 
a oe, oo: Se 
Lig | o 
Ht 4+-S00+ 
—100 | 130— 26 


| 26 cts. Ans. 


We will solve the above example, agreeable to the 
rules in common arithmetic. 


First, We are required to reduce compound fractions 
to simple ones. Rule, Multiply all the numerators together 
for a new numerator, and all the denominators together 
for anew denominator; then reduce all of these to their 
lowest terms. 


Reduce # of 3 of & of 74, to asimple fraction, 


Reduce 14, of 3 of 4 of § to a simple fraction. 


Tk Se dere 
4D, 7. OMS Fowl 


Reduce 5%; of § of 3 of 3; to a simple fraction. 
9 


= $10 9 


x 6 
Tio 8760 64 


x 
Ix bx é 


on 
xXx 


PROFIT AND LOSS. 123 


Then, if 4 of a yard cost 42; £, what will ~; of a yard 
cost ? 


i potest ie Salis Bhat 2 ana ‘ 
P.% ¥  AoAy oad 


6 Aa leh et 


fy 3, tire aad es shill. 1—6x12—18 
$...d. 
Wn ax 12-—90 11 of a $ 
cts. cts, 
100.2420..3-3) 130 
130 
1,00)26,00 


26 Ans. as before. 

8. A gentleman purchased a cask of wine, containing 220 
gallons, at 75 cents per gallon; he allows his agent 10 
per cent. commission for purchasing, and pays in gold, 
for which he is allowed a premium of 4 per cent., and by 
accident, 20 gallons leak out : how must the remainder be 
sold per gallon, to gain 20 per cent. ? 


920. 24 
15,3 

—110.\,100—+ 

—100 | 104— 26 


5 —200 
—4 —100 | 120— 6 


5 | 468—cts. 932 Ans, 


9. Bought 90 gallons of wine, at $1,20 per gallon, but by 
accident 10 gallons leaked out; at what rate must I sell 
the remainder per gallon, to gain upon the whole prime 
cost at the rate of 12 4 per cent. ? Ans. $1,51,8+-. _ 

10. By selling broadcloth at three dollars twenty-five 
cents per yard I lose at the rate of twenty per cent, ; 
what is the cost of the cloth per yard ? 

Ans. $4 06,25. 

11. If forty pounds of chocolate be sold at twenty-five 
cts. per pound, and I gain nine per cent., what did the 
whole cost me? — Ans, $9 17,4+. 


124 PROFIT AND LOSS. 


12. If I sell cloth at 5s per yard, and thereby gain 
fifteen per cent., what shall I gain per cent. if I sell it at 
6s per yard? Ans. 38 per cent. 

13. If I retail oil at one. dollar fifty cts. per gallon, and 
thereby gain twenty-five per cent., what shall I gain or 
lose per cent., if I sell it at one dollar eight cents per gal- 
lon? Ans. lose 10 per cent. 

14. If I sell one hundred pounds of sugar for eight dol- 
lars, and thereby lose twelve per cent., what shall I gain 
or lose per cent., if I sell four hundred pounds of the 
same sugar for thirty-six dollars? Ans. lose 1 per cent. 

15. A man sold a horse for one hundred and twenty 
dollars, and thereby lost twenty per cent., whereas he 
ought to have gained thirty per cent.; how much was he 
sold under his real value ? Ans. $75. 

16. If I buy six cwt. of sugar at ten pence N. Y. cur- 
rency per lb., and am allowed four per cent. discount for 
ready money, and sell the same so as to gain fifteen per 
cent. on the money advanced, how much money do I 
receive ? Ans. $77,28. 

17. Bought twelve chests of tea, each weighing fifty-six 
pounds, at four shillings and six pence N. E. currency 
per pound: for ready money was allowed two per cent. 
discount ; how much must I receive for the whole to real- 
ize a profit of 10 per cent. on the cash paid out ? 

Ans. $543,312 

18. How must cloth which costs thirteen shillings and 
four pence be sold to gain 121 per cent. ? 

Ans. 16 shillings per yard. 

19. Bought 1250 barrels of flour for $6250, at what 
price per barrel must I sell it to gain 124 per cent. ? 

Ans, $5,624. 
_ 20. Bought thirty hogsheads of molasses at $20 per 
hogshead, in Havana; paid duties $20,66, freight $40,78, 
portage $6,05, insurance $30,84; what per cent. should I 
gain by selling at $26 per hogshead? Ans. 11 5%935.+-. 

21. Bought wheat at seventy-five cents per bushel; at 
what rate must I sell it to gain twenty per cent. ¢ 

i Ans. 90 cts, 

22. If I purchase thirteen cwt. of coffee at 125 cents 
per pound, at what price per pound must I sell it to gain 
$80,80 on the whole ? Ans. 18 cts, 


PROFIT AND LOSS. 125 


23. A miller sold a quantity of rye at $1 per bushel and 
gained twenty per cent.; soon after, he sold of the same 
to the amount of $37,50 and gained fifty per cent.; how 
many bushels were there in the last parcel, and at what 
did he sell it per bushel ? Ans. 30 bushels, at $1,25. 

24. A trader bought one hogshead of gin, of a certain 
proof, containing 115 gallons, at. $1,10 per gallon; how 
many gallons of water must he put into it to gain $5, by 
selling it at $1 per gallon? Ans. 165 gallons. 

25. Bought four hogsheads of wine, containing 450 gal- 
lons, at $1 per gallon, and sold it at $1,20 per gallon, and 
gave 3 months’ credit; allowing the leakage of the wine 
while in my possession to be ten gallons, | would know 
the gain or loss, discounting for the present worth of the 
debt, at six percent. per annum. Ans. $70,19 gain. 

26. A yinter buys 596 gallons of wine, at 6 shillings 3 
pence per gallon, in ready money, and sells it immediately 
at. 6 shillings 9 pence per gallon, payable in 3 months ; 
how much is his gain or loss, supposing he allows the 
interest for the time at 6 per cent. per annum as discount 
for present payment? Ans. gain £11 17s. 8d. 

27.. What would be the gain er loss on the aforesaid 
wine, supposing the discount for present payment to be 
made at 2 per cent., without any regard to time? 

Ans. gain £10 17s. 63. 

28. A merchant bought a parcel of cloth at the rate of 
$1, for every 2 yards of which he sold a certain quantity 
at the rate of $3, for every 5 yards, and then found he had 
gained as much as 18 yards cost; how many yards did he 
sell ? Ans. 90 yards. 

29. A distiller is about purchasing 10,000 gallons of 
molasses, which he can have at 48 cents per gallon in 
ready money, or fifty cents per gallon with 2 months’ 
credit. It is required to know which is more advanta- 
geous to him, either to buy it on credit, or to borrow the 
money at 8 per cent. per annum, to pay the cash price. . 

Ans. he will gain $136 by paying cash. 


126. TARE AND TRET. 


“TARE AND TRET. 


Tare and Tret allowances made in selling goods by 
weight. 

Draft is an allowance ¢ on the gross weight in favor of 
the buyer or importer ; it is always deducted before the 
tare. 

Tare is an allowance made to the buyer for the weight 
of the hogshead, barrel or box or bag, &c., containing the 
commodity. 

Gross weight is the whole weight of the goods, together 
with the hogshead, barrel, or bag that contains them. 

Suttle is when part of the allowances is cece from 
the gross. 

Net weight is what remains after all ariaiah des are 
made. 

Tret is an allowance of 4 pounds for every 104 pounds, 
made for the dust, &c. 


Rule. Place the whole gross weight first on the right 
hand side of the line, then place 112 Ib =cwt. on the left 
hand side, with 112 diminished by the tare per cwt. stand- 
ing directly opposite on the right. 


EXAMPLES. 


1.. What is the net weight of 44 cwt. gross, if 14 Ibs. per 
cwt. be allowed for tare? 


—112 
2 —28 


44— J} 
98— 7 


2 77=384 ewt. Ans. 


2. Bought 84 ewt. of sugar; what isthe net weight, if 20 
Ibs. per cwt. be allowed for tare q 
—4 —16 —112 | 84— 12— 3 _ 
92— 23 


a a i es 


| 69 cwt. Ans. 


TARE AND TRET, 127 


3. Bought 7 hogsheads of sugar, each weighing 8 cwt. 
2 qrs.; from this a deduction of 16 lbs. per cwt. was made 
for tare; what was the net weight 1 


_ 8 
_— a 
ne. fe 7 met Ae 
a —-]6 —112 | 96—~ 48— 3. — 
sae 2 
51 cwt. Ans. 


4. What is the value of 8 hogsheads of sugar, each 
weighing 12 cwt., gr. tare, 12 lbs. per cwt. at $8, 40} per cwt.? 
12 
son] F194 100 
| 840— 60 


$720 
5. Bought 15 cwt. of sugar, at $6,50 per cwt, net 
weight, reduction for tare 12 lbs. per cwt., tret 4-Ibs. per 
{04 lbs.; how must I sell the whole to gain 20 per cent. 


_ on the foie cost ? 


15 
| 650— 325 
7 —28 —112 | 100— 25 
26 —52 —104 | 100— 
—100 | 120— s0— 15 
182 ! 182 8125 —$100,444+ Ans. 
6. Bought 742 lbs. gro. weight of cotton, and was allowed 
a deduction of 5 per cent. for tret, and for the net weight I 
paid 9 shillings (N. J. currency) per lb., and was allowed 
a deduction of 6 per cent. on the whole cost for ready 
money, I then sold the same so as to realize a profit of 20 
per cent. on the cash Ladvanced ; how much didI receive 
forthe whole ? 


ub kk ot 106is9 40 owas 
| 100— 2—0 
ODO: Ge oiZau. 
me are) 
—106 | 100— 
100.4 12 0 


—— $960 Ans. 


128 TARE AND TRET. 


7. A tobacconist buys 4 hogsheads of tobacco, weighing 
38 cwt. 2 qrs. 8 Ibs., gross, tare 94 lbs. per hogshead, at 
39 per cwt., ready money, and sells it at 115 pence per 
Ib. , allowing tare at 14 Ibs. per ewt., to receive 2 in cash, 
and for the rem inder a note at 90 days’ credit ; * his gain 
or loss is regi , Supposing the interest for the time at 
6 per cent. per 360 days 1s allowed for discount, on turn- 
ing the note into cash. Ans, $283, 80 gain. 


8. Bought 32 casks of figs, each weighing 2 cwt. 2 qrs. 
at a deduction of 18 lbs. per cwt. for tare, what did the 


whole cost me, at $4 per cwt. net weight ? 
Ans. $268,57. 


9. Bought 32 chests of tea, each weighing 4 cwt. 2 qrs. 
at $49 per ewt. net weight, tare 12 lbs. per cwt., tret 4 
Ibs. per 104 lbs., how must I sell the whole quantity to 
gain 20 per cent. ? Ans. $7269 23. 


10. Bought 5 cwt. of sugar, tare allowed 8 lbs. per cwt., 
for the net weight I paid 6 pence (N.Y. currency) per 
pound; how must I sell the whole quantity to gain 20 per. 
cent. ? Ans, $39, 


11. Purchased 12 bain of coffee, each weighing 96 lbs., 
tare per bag 6 lbs., what was the whole cost at 30 cents 
per lb., and the retail price to gain 25 per cent. ? 

Ans. Cost $324, retail price 374 cts. 


12. How much will 8 hogsheads of sugar, each weigh: 
ing 8 cwt. 3 qrs, cost at $9 per cwt., if a deduction of 12 
lbs. per cwt. be allowed for tare, and what will be received 
for the whole, if it be sold at an advance of 30 per cent. ? 

Ans. Cost $562,50, received $731,25. 


13. What is the net weight of 3 tierces of rice, each 
weighing 4 cwt.3 qrs. gross, tare 16 lb. per cwt., tret 4 
los. per 104 lbs. Ans. 11 cwt. 2 qrs. 27 Ibs s.. 


14. Purchased in London 16 cwt. of tea, at £28 ster- 
ling per cwt., net weight, tare 12 lbs. per cwt., how 
much must I receive, in federal money, for the whole 
quantity, to realize a profit of 12 per cent., and what 
retail price will allow the same profit ? ) 

Ans. Wholesale $1991,11, retail $1,24. 


15. Bought 16 firkins of butter, each weighing 108 
Ibs , reduction for tare 8 Ib. per cwt., paid 15 pence New 


C@MMERCIAL EXCHANGE. «~ 129 


England currency per lb., what did it cost ‘me, and what 
must be the wholesale ee retail price, to ean 2m per 
cent. on the first cost 2 ak 

Ans. Cost $334,281 wholesale $401,141 re 


Under this rule are included the operations of purchas- 
ing goods in one country and selling them in the currency 
of another country, 80 as to gain or lose some required 
per cent. 

Rule. Place the whole cost in the given currency, first 
on the right hand side of the line, (if the retail price be 
required) the number expressing the quantity procured for 
that price first on the left hand side, write next on the 
right the value of a unit of the given currency, in federal 
money, and lastly to increase or diminish the price by the 
required per cent; place 100 on the left of the line, and 
100 increased by the per cent, to be gained, or diminished 
by the per cent. to be lost on the right hand side of the 
line. 


ote. If the wholesale price be required, the number 
expressing the whole quantity (by the preceding rule 
placed on the left of the line) must be rejected. 


EXAMPLES. 


1. Purchased in London 360 yards of Ropers which 
cost me, including transportation, £300, sterling, how 


must I “ape e same per yard, in federal money, to make 
per cent. 4 


a, profit of 
3002— 


9 | 40=$4,444 Ans. 


12 


24 180P : COMMERCIAL EXCHANGE. 


ip ‘ul chased i in Sdiddhe 350 yards of sheeting, for £75, 
rats transportation to New York ; how 
ne, in federal money, to gain 15. ‘per 


ae 


40— 


—100} a 23 


ghd 75:14. 2001 —$1 327+ Ans. » 


3. A merchant in London bought 700 ells: of cloth, at 5 
i I eli, the cost of transportation and 


pount was 35 p per cent., the exchange 


’ a nes on the ; fer how many cents 
ust one yard be sold in Philadelphia, to gain 124 per 
cent? 
Yards. = = —5|4— English ells. 
Rae VAL Oep Bas eealig 
—100 | 100— 
135 


pad 100 | 225) Dien 
ae 


a r 


$1,35 Ans. 
Note A ioneuett who is not acquainted with this sys- 
te oe See, Berea by the Bonet 1 caleula- 


As proof, we will go a AD, the caleulk n of the six 
eee by the rule of three : | ae 

Ist Question. How many &s serlings ells of 
cloth cost, at 5 shillings per ell ? 


ell shill. sigan 
Oe Re ese kg pa wi 
700 pigs 


2(0)350(0 Ze ‘ 


£175 


= 


COMMERCIAL EXCHANGE. — 


2d. QRestion: ‘How much will LLT5 7 
per cent ? 


d f deostade: In 236} - pounds, how many $s at 41 2 
lings each? = 
=A shill: § ie + 


AMER es ABOY 4°72500(1050 
| 450 


$ 
ae 12, pu - 1050 
1050 


5 (Ser 
11250 


Cc om. MERCIAL Rite ek ° 


oth fenestions, Tn 700 ells how 1m many yards ¢ F e | 


If ‘819 yards =e t. $1181, ee? 


, 81 


. Receis rom ngland 3 hhds. of wine at 10 pence 
no per rat for which I paid for nd duty ab 

er cent., exchange being 10 per cent : 

man) dollars will 3 hhds. cent in Penn, 


ys teelQ) ‘ 40--- 
---100 | 125--- 5 
---4-=-=100 | - | 


me res aS 


fis? sum of e788 at 10 per cent. premiu 
$3828 or pay the same to a broker to « 
if New York funds are 6 per cent bet 
Philadelphia, and that a Phil 
ceived the same merchandise fr much 
must he pay the broker 
783- 
n-eQ | 40-.- ee 
I-60) £1s0—2 
25.100 + 106 


} New York me | 
: : 229,68 diff. 


5. A merchant had 18000 lbs. of wool, which he could 
sell at 9 pence, New England. currency per lb., but not 
finding a purchaser to suit him he be tered. with A and 
gave 74 lbs. of wool for two yards linen, not yet finding a 
cash customer, be bartered his linen with B for sugar, and 
gave 24 yards of linen for 34 lbs. of sugar, and changed his 
sugar with an Englishman for broadcloth, and gave 1 cwt. 
of sugar for 20 ells of broadcloth. . Now he s¢ sold. pia cloth 
for $2 per yard, what did he lose or gain: by this tr 
tion, estimating. the wool 9 pence (New Bogland cur- 


Rabe 
Det 
ie os 

|. 20-- 
rome 
9... 


| $3000. 


“750. b diarene, gain, 


oe % ae 


134 COMMERCIAL EXCHANGE. 


6. A gentleman of the United States left his 7 children 
a piece of land 6 miles long, 4% miles broad. A son set- 
tled in England, sold his part there for £44 sterling per 
acre, but being indebted to his relations he sent the money 
to the United States, exchange at that time 5 per cent pre- 
‘ mium. _For what ought the bill to draw? 


are ee Th 
“ee 6:-- 3 
---4 | 19 
64 Qu 
oon Dat eiaaill : 
229 |-40--2 


|. $54720. Ans. 


7... A Frenchman had failed in trade by misfortune, but 
his father being wealthy resolved to establish him again 
in the United States, and furnished part of the cargo ofa 
vessel, and took passage with his son in her; two sons of 
the father remained in France, also three children of his 
son, but two of them, brother and sister, had taken pas- 
sage ina packet ship. The vessel with father and:son 
was lost at sea. The cargo was insured for 1,347,840 
francs, of which the grandfather possessed 4, part: By 
chance a young clerk became acquainted with the brother 
and sister, his principal being appointed commissioner 
from the different partners of the insurance to settle the 
business in France, allowing him 4 per cent. commission. 
The clerk and brother having become intimatesfriends, he 
proposed to marry his sister, and to commence business 

for themselves by selling their parts of the iapbrnc to 
~ said commissioners, at a~ discount of ten per cent, and as 
they agreed so, the question is, with how much federal 
money did the brothers-in-law begin business, if exchange 
at that time was 104 pence sterling per franc, and from 
England to America 10 per cent. premium ? 


{Solution next page. | 


COMMERCIAL EXCHANCE. 135 


—5 
gui 
—3 —12 i 


FS 


42 
hee ri 


1347840— 149760— 
22 | 21 ye age 


—6 —12 | ~  8320— 
—20 | 80 
—9 | 40— | 
- —100 | 110~— 
—110 | 100— 
104— | 100— 5 
| $2800. Ans. 


Ta 


8. A merchant imported from England 975 ells of 
eloth at 7 shillmgs 6 pence per ell. The commission and 
duty amounted to 40 per cent., exchange 10 per cent. 
premium, how must 1 yard be sold inthe United States 
to gain 124 per cent ? 

For the proof state the example as before, then place 
the answer on the left, and if all the numbers cancel, the 
work 1s right, thus: 


---1] —231 -25 | Ae 
---5 | 4--- 222i 15 Bee 
~--2 | 15--- 3---  ---3.---9 | 2--- 
2629. | Qe~ ---100 | 100--- 
---100 | 100— | 140--- 7 
140----7—..:..---100, 4: 1b .0— 

+ ---100 | 110--- .. —4 —100 | 225-7513 
---25 —100 | 225— 9-- —2.1 ‘ | 
ec | ae 

teas | $2,341 


9. Received from London 470 yards of linen, which, 
including transportation, cost me 65, sold the same by 
the yard. so as to gain 30 per cent. on the first cost, how 
did I sell it 2 Ans. $, 79+. 

10. Received from Dublin 600 yards of Irish linen, the 
whole cost of which was -€75 Irish currency, how must I 
retail the same in federal money to gain 124 per cent ? 

Ans. $0,576 per yard. 


136 COMMERCIAL EXCHANGE. 


11. Received from my agent in Dublin 900 yards of 
linen, whole cost £60 Irish currency, how must I retail 
the same in federal moniey;to gain 15 per cent ? 

Ans. $0,31 Ate 

12. I have in my store 120 yards of broadcloth for- 
warded me by my agent in Paris, which cost me including 
transportation 325 crowns, how must I sell the same in 
federal money, to gain 16 per cent. ? 

Ans. $3,455-+- per yard. 

13. Received from Madrid 6 hhds. of wine, each con- 
taining 63 gallons, for which my agent paid 188 Spanish 
dollars, how must I sell the same per gallon to gain 124 
per cent ? Ans. $0,559-+, 

14. I have on hand a balg of silk containing 174 
yards, which I received from Cadiz, at a cost, including 
transportation, of 140 piasters or Spanish dollars, how 
must I sell the same per yard to gain 6 per cent. 4 

Ans. $0,844. 

15. Received from Oporto 3 hhds. of port wine, contain- 
ing 63 gallons each, cost including transportation, 30 mil- 
rees per hhd., how must. I retail the same per gallon to 
gain 25 per cent. ? Ans. $0,738->e 

16. Consigned to my agent, J. Smith, of London, 300 
barrels of flour, for which I paid $1500. How many L's 
sterling ought he to receive for the same to gain 10 per 
cent., the expense of transportation being $50. 

Ans. £383 12 shillings 6 pence. 

17. Received of my agent in London, J. Smith, 
2510 gallons of Madeira wine, which cost me per invoice 
£1640 sterling, buat it being of an inferior quality I am 
willing to lose 5 per cent. on the cost, what must be the 
ptice per gallon in federal money ? Ans. $2,758-+. 

18. Three men traded in company, received from 
France 1200 bottles of champagne, for which they paid 
600 French guineas, each $4,60, how must they sell the 
same per bottle in federal money to gain 40 per cent., and 
what will be each man’s gain per bottle ? 

Ans. $3,22 each man’s gain, $0, 306-++. 

19. Received 300 ells of cloth from Hamburgh, which 
cost me 1500 mark bancos, how must the same be sold in 
federal money by the yard to gain 124 per cent , the ell of 
Hamburgh being 24 qrs. Ans. $1,17-+. 


UNCOINED AND SILVER MONEY. 137. 


20. New Yorn, June 9th, 1840. This day, received 
from Amsterdam 600 yards of carpeting, whole cost 2400 
guilders, required the retail price in federal money to 
gain 20 per cent ? Ans. $1,92. 

21. Shipped to London 380: barrels of flour, which cost 
me, including transportation, $6 per barrel, how many 
English crowns must I receive for the whole quantity to 
gain 10 per cent ? Ans. 2280 crowns. 


TABLE OF UNCOINED AND SILVER MONEY. 


ENGLISH CURRENCY. 


£1 sterling (before 1832) . - ~» equalled $4,44 
£1 “«”, (since 1832) - - - ‘§ 4,80 
Or prior to 1832 £9 - - - «40,00 
Since 1832 £5 - > - - equals 24,00 
1 English crown - - . - Af 1,10 
or 10 English crowns” - - Le Hh ed aD 
1 English shilling - - - - ee 224 
1 pistoreen - - - - - ss 20 
1 English penny ° - - - “01% 
IRISH CURRENCY 

£1 Irish - . - - ~ - equals $4,10 
or £10 nS rare Hela emits - ye vo 41,00 
1 shilling t= : - - . ae 2c; 
1 penny - - - - = = Ore 

re wa) 


FRENCH CURRENCY. 


1 French Crown - - Se equals $1,10 
or 10 crowns - cer eer - - ‘ 11,00 
1 five franc piece - - - rides 593 
1 frane an - - - - ss 182 
1 Decimes - - - - - ef 01,26 


SPANISH CURRENCY. 


1 Spanish dollar - - - - danas $1,00 
1 real newplate - - - - ,10 
1 real vellon - - : - 4 05 


138 SINGLE FELLOWSHIP. 


CURRENCIES OF OTHER NATIONS, 


1 millree of Portugal - - - equals $1,24 
1 Russian silver ruble - - - Sian a 75 
1 rix dollar of Sweden - - - “ 1,00 
1 Russian. rix dollar - - . ot ,663 
or three rix dollars - - - = “2°00 
1 Danish rigsbank dollar nt 2 Saer tae ,50 
1 silver ducat of Naples - - - As 80 
ot 5 ducats °- vise - - - se 4,00 
1 scudo of Sicily - : ores ‘6 ,96 
1 oncia of Sicily - - : ‘ . “« 240 
: pizza of Leghorn - - - - é ,90 
1 pizza ofGenoa~ - - . - . a 589 
1 florin of Trieste - - - E és 548 
1 Rix dollar of Trieste - - - - ‘“ ,96 
1 Roman Crown . : - - “? "500 
1 gold Crown of Rome - - - 5p 5S 
1 Maltese scudo - - - af cs ,40 
1 rupee of Bengal. - - - “ : és 552 
1 rupee of Bombay - - $d és , 
1 pagoda of Madrass - - - - “« 180 
1 tale of Canton - - ~ P coe eas 
1 Japanese tale - - : a Line 75 
1 dollar of Sumatra - - - - « a dgkO 
i tale of Sumatra - - - - - co  A16 
1 florin of Java - - - “ ts ,40 
1 mark banco Bitamburah - a . gaa 
1 guilder of Amsterdam - - e és ,40 


SINGLE FE LLOWSHIP. 


Rule. Place the amount which each partner put in on 
the left hand side of the line, and the whole gain or loss, 
together with the stock each partner put in severally, on 
the right hand side. 


8 
EXAMPLES. 


Three gentlemen, A, B and C, shipped a number of 
horses to the West Indies, of which A owned 24, B 36, 
and © 48, and in distress of weather the seamen were 


SINGLE FELLOWSHIP. 139 


obliged to throw 45 overboard, what part of the loss does 
each sustain. 


A owned 24 
i: eae 36 Whole loss 45. 
Cree 48 


Whole stock 108 


—9. 108 |2te~ 2 43) | 368 9148 4 
[ode A. ee b08 | 45S bS) eh OR FAS 


Se ees, 


———— @ re 


| 10 A’s, 115 B's, | 20 C's. 


45 proof. 


A, Band C entered into partnership, A invested $400, 
Ts $600, and C $1000, and they gained $800, what is each 


Gucs suare of the gain. : ~ 400 
600 
1000 
2000 
--5 ---2000 | 400— —2000 | 600— 3--2000 | 1600— 
| 800— | 80 —O | $800— 400 
| $160 A’s. | $240 B’s. | $400 C’s. 


3. Two persons hired a coach in Boston to go 40 miles 
for $20, with liberty to take im two-more when they 
pleased. When they had gone 15 miles they admit C 
who wished to go the same route and return with them to 
Boston, and on their return, within 25 miles of Boston, 
they admit D for the remainder of the journey. As each 
person is to pay in proportion to the distance he rode, it 
is required to settle the coach hire between them. 


250 whole distance. 


140 SINGLE FELLOWSHIP. 


8 o— | 8 0— 
5 2 R50 war 5 —250 
| 20— 4 20— 4 
5 | 82=$62 5 1 32=$62 
| 65— 13 | 25— 
5 —250 } 250. | 
| 2 —0 ) 2-0— 
5 | 26=$52 | $2 
A’s 62 
B’s 62 
C’s 51 
D’s 2 
$20 


4, A Seutomae distributed $60 among 4 of his servants, 
giving to A 4, to B 4, to C4, and to D 4, what. vach 
ones share? ‘ . 


+ of 6020 
1 “ 60=15 
1 « 60=12 . 
i « 60=10 
by aes 
20 15 
—d7 19 —d7 
19 | G60O— 20 60— 20 
19 | 400=$21,3, 19 | 300=$1518. 
12 ) 10° 
19—57 19.—57 
60— 20 60— 20 
19 | 240=$1212,.. 19 | 200=$1022. 
A’s 21,}5 
B’s 1542. 
C 1242 
D 10° 


$60 proof. 


COMPOUND FELLOWSHIP. 141 


5. Six gentlemen, A, B, C, D, E and F entered into 
partnership for one year; A put in two hundred dollars, 
B three hundred, C four hundred, D five hundred, E six 
hundred, F eight hundred, and they gained five hundred 
and sixty dollars; what was each one’s share of the gain ? 

Ans. A’s share 40 dolls.; B’s 60, C’s 80, D’s 100, E’s 
120 and F’s 160. 

6. Divide the number 360 into three parts which shall 
be to each other as 2, 3 and 4. Ans. 80, 120 and 160. 

7. Two merchants have gained £450, of which A is to 
have three times as much as B; how much is each to 
have ? Ans. A’s share £337 10s., B’s £112 10s. 

8. Three persons are to share £600; A is to have a 
certain sum, B as much again as A, and C three times as 
much as B; what is each part ? 

Ans. A’s £662, B’s €1334, and C’s £400. 


ES SRE SS AR SA SE RN SDE TRS ee 


COMPOUND FELLOWSHIP. 


_ Rule. Multiply each man’s stock by the time during 
which he continued in trade, then place the amount of the 
several products on the left hand side of the line, and the 
gain or loss, together with the product of the stock of 
each of the partners, severally multiplied by the time, 
on the right hand side. 


EXAMPLES. 


_ 1. A, B, and C hold a pasture in common, for which they 
pay $56. In this pasture A has 40 sheep for 6 weeks, B 
60 sheep for 8 weeks, and C 80 sheep for 12 weeks. What 
part of the $56 ought each to pay ? 


A 40X 6=240 
B 60X 8=480 
C 80X12—960 


* 1680 whole stock, 
[Solution opposite page.] | 


142 COMPOUND FELLOWSHIP. 


240—.8 480— 16 
—3 —1680 | 56— —3 —1680 | 56— 
| $8 A’s. | $16 B's. 
8 
16 | 960— 32 
29, ior eB: T6B0 |. 56 
$56 proof. $32 C’s. 


2. Three merchants entered into partnership ; A put in 
200 dollars for two months, B putin 400 for three months, 
and C put in 200 for 7 months; they gained 600 dollars ; 
what is each’s share ? 

Ans. A’s share $80, B’s 240, C’s 280. 


3. A, B,C, D, E, F, Gand H entered into partner- 


ship ; A’sstock was three hundred dollars for four months, — 


B’s four -hundred for six months, C’s five hundred for 
eight months, D’s six hundred for five months, E’s eight 
hundred for three months, F’stwo hundred for five months, 
G’s five hundred tor two months, and H’s one thousand for 
five months ; they gained nine hundred and sixty dollars ; 
required each one’s share of the gain. 


4. Two merchants traded in company, A put in 215 dol- 
lars for 6 months, and B 390 dollars for,9 months, but by 
misfortune they lose two hundred, how must they share 
the loss ? Ans. A’s $53,75, B’s $146, 25. 

5. Three persons had received 665 dollars interest; A 
put in four thousand dollars for 12 months, B three thou- 
sand for 15 months, and C five thousand for 8 months; 
how much is each man’s part of the interest ? 

Ans. A’s $240, B’s 225, C’s 200. 

6. A and B companied; A put in two thousand dollars 
Jan. 1, but B put in his share June 1; what did he then 
put in to have an equal share in the profits with A ? 

Ans. $34284. 

7. Three merchants traded in. company ; A Pat in one 
hundred and twenty dollars for ten months, B one hun- 


dred for 18 months, and O one hundred fifty f for’ months ; 


~ 


MENSURATION. 143 


they gained one hundred dollars ; what was each man’s 
share ? Ans, A’s $32, B’s 48, C’s 20. 
8. E andS enter into partnership for 1 year; E first 
advances. four hundred eighty dollars and B puts in his 
share 3 months after ; how much must he advance to be 
entitled to an equal share of the gain at the expiration of 
one year ? Ans. $640. 
§. T'wo merchants trading in company gain two hun- 
dred dollars; A’s stock was two hundred and twenty dol-. 
lars for 6 months, and B’s 380 dollars for nine months ; 
how ought they to share the gain ? 
| Ans. A’s part $55,69,6, B’s $144, 30, 4. 
10. Two men commenced trading in company Jan. 1, 
1841; A advanced one thousand dollars, at the time 
specified, but B did not advance his share till the first of 
May following; at the end of the year they shared the 
profits equally ; what capital did B advance ? 
_ Ans. $1500 


MENSURATION 
OR 
PRACTICAL GEOMETRY. 
BOARD AND TIMBER MEASURE. 


Rule. For measuring boards, place the length in feet 
and. width in inches on the right hand side of the line, and 
12 onthe left hand side. But for square timber, place the 
length in feet and the width and thickness in inches on the 
right hand side of the line, and 12 for board measure or 


144 for cubic feet on the left of the line. . 


EXAMPLES. 


1. How many feet. in a board 48 feet long and 13 inches 
wide ? 
48— 4 
. ; —12} 13 


| 52 feet Ans. 


144 MENSURATION. 


2. How many feet in a board 40 feet long and 27 inches 
wide? ) 
a—— <0 
—3 —12 | 27— 9 


| 90 feet. Ans. 


3. How many feet in a board 6,5; feet long and 33 inches 
wide ? | 
—l1l1 | 72— 6 
—12 | 33— 3 


—ae 


| 18 feet.. Ans. 


4. How many feet in a board or plank 44 feet long and 
74 inches wide ? 


—5 | 24— 
atk es Pee 
—12 

| 3 feet. Ans. 


5. How many feet of boards ina square stick of timber 
(making no allowance for the saw) 60 feet long 7 inches 
wide and 6 inches thick ? 

60— 5 
7 
6 


—12 


| 210 feet.. Ans. 


6. How many feet of boards in a square stick of tim- 
ber 75 feet long, 18 inches wide and 8 inches thick ? 


TS SEE Cr TS 


| 900 feet. Ans, - 


MENSURATION. 145 
7. How many cubic feet in a stick of square timber 96 
feet long, 24 inches thick, and 37 inches wide? 
96— 16 
—6 —144 | 24— 
ore 


re connec 


1 092 feet. Ans. 


8. How many cubic feet in a square marble pillar 142 
feet long, 15 inches wide, and 8 inches thick ? 


—5 | 72— 
—2 —144 | 15— 3 
| 8— 4 


i es ee, 


i 12 cubic ee. Ans, 


9, How many ohne feet i in a stone or stick of timber 
108 feet long, 44 inches Wide and 3} inches thick ? 


---6) 2244 } 108.-16 
Se yy 
wa Gain 


| 12 cubic feet. Ans. 


10. What is the content of a piece of timber 40 feet and 
the sides 18 by 21 inches ? Ans. 105 feet 


11. What is the content of a piece of timber 48 feet long, 
and the sides 14 by 9 inches ? 


—~ 


‘| 42 cubic feet. Ans. 


12. What is ‘the content of a piece of timber 9 feet long 
and the sides 32 by 7 inches? 


Ans. 14 feet. 
13. How many cubic feet in a stone 20 feet long, 36 


inches wide, and 4 inches thick ? 
Ans. 20 feet. 


14 


146 MENSURATION, 


To find the solidity of a cylinder. 


Rule. Place the square of the diameter, the decimal, 
7854* and altitude, on the right hand side of the line, and 
the denomination next inferior to the answer on the left. 


EXAMPLES. 


1. Required the solidity of a cylinder the diameter 
36 inches, and the length 20 feet. 
36--- 
re 
---4 ---144 | »7854 
| 20--- 5 


| 141, 372 feet. Ans. 


2. Required the solidity of a cylinder, the diameter 
being 4 44 inches, and length 50 feet ? 


---) | 24--- 4 

Ee ae 

---6 ~--144 | ,'7854 
1 O0--- 2 


ES 


| 6,2832 feet. - Ans. 


3. Required the solidity of a cylinder the diameter 
being 9 inches. and altitude 12 feet. Ans. 5, 3014. 
4, Required the solid feet contained in a stick of timber 
of equal thickness, the diameter being 9 inches, and length 
24 feet. Ans. 10, 6029. 
5. Required the cubic feet contained in a round stick of 
timber of equal bigness from end to end, the diameter 
being 18 inches, and length 36 feet. Ans. 63, 6174. 


To square round timber. 


Rule. Place twice the square of the semi diameter, . to- 
gether with the length on the right hand side of the line, 
and the denomination next inferior to that in which you 
wish your answer on the left. 


* As the area ofa circle whose diameter is 1 inch, is, ,7854 deci- 
mal parts of an inch. 


MENSURATION, 147 


EXAMPLES. 


1. What will be the solid feet of a round stick of tim- 
ber 24 inches diameter and 40 feet long, when hewn 
square % | 

| 12--- 
12--- 
2 
1 40 


| 80 feet. Ans. 


---144 


Norse. The square of a number is that number multi- 
plied by itself. 


2. What will be the solid content of a round stick of: 
timber when hewn square, of 44 inches diameter and 75 
feet long ? 


---5 | 24--- 4 
---D | 24--- 
---G ---144 | 2 
1 75--- 3 


| | 24 feet, Ans. 


3. What will be the solid content of a round stick of 
timber, when hewn square, of 5 feet diameter and 48 feet 
long 4 ii 


60 

60--- 20 
2 

48--- 


—_—_ SS 2 - _ 


| 2400°feet. Ans. 


---3 ---144 


GAUGING, 
To find the number of gallons contained in a circular cistern. 


Rule. Place the square of the diameter, the length, and 
47 on the right hand side of the line, and 8 on the left, for 
gallons. 


148 MENSURATION. 


EXAMPLES. 


1. Required the number of gallons in a circular cis- 
‘tern, the diameter being 8 feet and height 12 feet? 
8 
---8 | 12 
1 47 


ss 


| 4512. Ans. 


2. Required the number of hhds. in a circular cistern 
16 feet diameter, and 20 feet deep, allowing the hhd. to 
contain 160 gallons. 


3 hers 
eg Ms 
goo bo Oe 

47 


| 188hbhds. Ans. 


3. Required the number of hhds. in a circular cistern 
44 feet diameter, and 33} feet deep, allowing the hhds. to 
contain 141 gallons ? 


sikh | SP ce 
yy ere 
3 | 100--- 4 
Te a 
03> xcedi | 
| 32 Ans 


4, Required the number of gallons contained in a circu- 
lar cistern, the diameter being 8 feet and the depth 4 feet ? 
Ans. 1504, 


5. Required the number 3 gallons contained in a cir- 


cular cistern of 12 feet diameter and 9 feet in depth ? 
Ans. 7614. 


6. Required the number of gallons contained in a cir- 


cular cistern, the radius being 8 feet and height 10 feet ? 
Ans. 15040. 


MENSURATION. 149 


Nore. An ale gallon contains 282 cubic inches. 


A wine gallon - - 231 
A bushel - - He 2150 
A cubic foot - J - ¥728 


The als gallon is to the wine gallon as 58 to 7! nearly. 


To find the contents of a cask. 


Rule. To twice the square of the bulge diameter add 
once the square of the head diameter, and place this 
sum on the right hand side of the line, together with the 
length of the cask, then place 1077 for beer or 882 for 
wine on the left hand side. 


EXAMPLES. 


1. What is the content of a cask whose bulge diameter 
is 40 inches, the head diameter 30 inches, and the length 
60 inches, in wine measure 2 


40 30 
40 30 
1600 9006 
I, ~ 
3200 
7 900 
4100 
“889 | 60. sae 


147 —294 


eee 


147 | 41000278134 galls, Ans. 
2. What is the content of a.cask in beer gallon whose 
bulge diameter is 50 inches, the head diameter 36 inches, 
and length 90 inches ? 


[Solution, next page. | 


150 MENSURATION, 
50 
50 36 
“ —- 36. 
2500 —- 
2 216 
—- 108 < 
5000 eee 
1296 1296 
| 6296 
—1077 | 90— 30 
359 


359 | 18888052645, galls. Ans. 


Seaeenreeemaaeneamdl 


To find the contents of a figure that has six sides, and the 
opposite sides parallel. 


Rule. Place the length, breadth and apne on the right 
hand side of the line, then place on the left 2150 for bush- 
els, or 282 for ale gallons, or 231 for wine gallons, or 1728 
for solid feet. 

EXAMPLES. 


1. How many bushels of wheat will a box contain of 
the following dimensions ; 86 inches long, 71 inches deep, 
and 25 inches wide 2 


86— 


Fah 
25— 


—86 —2150 


71 bushels, Ans. 
2. Hoe many bushels of corn will a box of the following 
dimensions contain ; 645 inches long, 55 inches deep, and 
22 inches wide ? 


—2 | 645— 3 
—2150 | 55— 11 
22— 11 


ere ee 


| 363 bush. Ans. 


a ee 


‘i MENSURATION. 151 


3. How many bushels of rye will a box contain; 33} 
inches long, 17,2; inches wide, and 61 inches deep ? 


S100 3 
1 179 2 
Pa 
—86 —I150"| 


—eee 


3 |* 5=12 bushels, Ans. 


4, Ifthe length of a vat be 70 inches, breadth 163 inches, 
and depth 47 inches, what will be the content in wine 
gallons ? 


| 70— 5 
4-81 83t = 


——33 —231.|,47 


|. 235 gallons, Ans. 


5. If the length of a vat be ten feet six inches long,. 
five feet six inches deep, and three feet eleven inches 
wide, how many beer gallons will it contain, and hhds., at 
63 gallons each ! 


126— 2 ; 
+-§ = 889166 
soe | 


er 


| 22 hhds. or 1386 galls. Ans. 


To find the burthen of ships. 


Rule. Place the length of the keel in feet, the breadth 
of the mid-ship-beam, and the depth of the hold, on the 
right hand side of the lite; and 95 on the left for mer- 
chant ships, but for ships of war place on the left 100, and 
the answer will be in tons. 


152 MENSURATION. 


| EXAMPLES, 


1. What is the tonage of a merchant ship, length of 
keel 250 feet, depth of hold 11 feet, and breadth of beam 
19 feet 2 fe 
250— 50 


~-5 —95 | 19— 
rece! 
sl 


| 550 tons, Ans. 


2. What isthe tonage of a merchant ship, length of 
keel 275 feet, depth of hold 12 feet 8 inches, breadth of 
beam 27 feet ? 


—19 —95 | 275— 55 


| 990 tons Ans. 

_3. What is the tonage of a ship of war, length of keel 
260 feet, breadth of beam 25 feet, depth of hold twelve 
feet 4 ier 

| 260 
—4 —100 | 25— 
| Leaman 


eee 


| 780 tons, Ans. 


4, The. proportions of Noah’s ark, (Gen. vi. 15,) were 
as follows: length 300 cubits, breadth 50 cubits, and depth 
of the hold 30 cubits. Require its burthen, allowing the 
cubit to be 22 inches ? 

Ans. 29188,-+as a merchant ship. 
27729,+-“ “ ship of war. 


To find the area of a square, a rectangle,a rhombus, or a 
parallelogram. 


Rule.” Place the base and perpendicular heighth on the 
right hand side of the line, and the denomination next 
inferior to that in which we wish our answer on the left 


hand side. 


“ MENSURATION. 153 


EXAMPLES. 


1. Required the area of a square piece of land 80 rods 
_#quare. 

80— 40 
—2 —160 | 80— 


| 40 acres, Ans. 


2. Required the area of a square piece of land 480 rods 
square. 
: 480 | 
—160 | 480— 3 


| 1440 acres, Ans. 


3, Required the value of a piece of land 6,% rods 
square, at $12,10 per acre. . 


PP Woes 78 
Bs i iar SS 
SEPP re te 


| $3,24, Ans. 


4, Required the area of a parallelogram, whose length 
is 480 rods, and width 96 rods. 


480— 3 
96 


? 


—160 


| 288 acres, Ans. 


5. Required the number of acres in the road from New 
York to Philadelphia, the distance being 96 miles, and the 
average width 4 rods. 


96 

8 er 
40— 
ban 


—160 


| 768 acres, Ans. 
15* 


154 MENSURATION. 
co 
6. How many acres in a square piece of land whose side 
is 32 rods ? Ans. 62 acres. 
7. Required the area of a square piece of land whose 
side is 15 chains ? Ans. 225 acres. 
8. How many men can stand on 5 acres of land, each" 
man occupying a space of 3 feet square ? 
Ans. 24,2 0 men. 
9. A gentleman purchased a farm in the form of a 
square, at 48 dollars per acre. Required the cost allowing 
the side to be 25 chains. Ans. 3000 dollars. 
10. Required the area of a parallelogram, whose. base 
is eighty rods and altitude 25 rods. Ans, 12 acres 2 rods. 
11. How many acres ina field in the form of a paral- 
lelogram, whose base is 95 rods, and altitude 40 rods ? 
Ans. 23 acres 3 roods. 
12. Required the area of a field in the form of a paral- 
lelogram, whose base is 35 chains and altitude ten chains. 
Ans. 85 acres. 
13. Four Sel aes purchased a farm in the form of a 
parallelogram, the base thereof was 320 rods and altitude 
90 rods, and divided it equally. Required the portion of 


each. Ans. 45 acres. 
14. Required the area of a rectangle, whose base is 28 
feet and breadth 9 inches. Ans. 21 feet. 


15. How many. square feet are there in a rectangular 
board, wanse length is 36 feet and breadth ten inches 2 


Ans, 30 feet. 
16. ioniined the area of a rectangular farm, whose 
base is 88 rods, and breadth forty rods. Ans, 22 acres. 


17. How many acres are there in a rectangular farm, 

whose base is ‘fifty chains, and breadth twenty chains? 
Ans. 100 acres. 

18. Required the area of a rhombus, whose base is 75 
rods, and breadth forty rods. Ans. 18 acres 3 roods. 

19. How many acres are there in a farm in the form of 
arhombus, whose base is forty-five chains, and breadth 
twenty chains ? Ans. 90 acres. 

20, Required the area of a rectangular board, whose 
length is twenty feet, and breadth one foot four inches, 

Ans. 262 feet, 

21. Required the area of a parallelogram whose eer is 

eight hundred rods, and altitude four hundred rods. 


MENSURATION. 156 


To find the area of a trisgle. 


Rule. Place the altitude and half the length of the base 
on the right hand side of the line, and the denomination 
next inferior to that corresponding with the answer on the 
left hand side. es 

Norr. A triangle is equal to half a parallelogram of the 
same base and altitude, therefore the truth of this rule is 
evident. 

EXAMPLES, 


1. Required the area of a triangle, the base ine eee 
rods and altitude forty rods. 
| | | 30— 15 
ik Dood eal GO FeO 


a a a ES 


2..| 15=273 acres, Ans, * 


2. Required the area of a triangle, the base being six 
hundred rods, and altitude eighty rods. 


| 300— 150 
2-160] 80 


|. 150.acres, Ans. 


3. Required the area of a triangle, the base beni 
ninety-eight rods, and altitude 614 rods. 


pagwey 
pop BBs OT 
= 1600" 


10 | 189=18,9, acres, Ans. 


* 


4, Required the area of a triangle, whose base is ninety 
rods, and altitude sixty rods. 
Ans. 16 acres, 3 roods, 20 rods. 
5. How many acres im a triangle, whose base is 120 
rods, and altitude eighty-four rods? Semi te 31 acres, 2 roods. 
6. A gentleman purchased a triangular farm, the base 
thereof was 480 rods, and altitude 120 rods. Required the 
cost of said farm at fifty dellars-per acre. Ans, £9000. 


156 MENSURATION. 


7. Required the value of a triangular farm, the base 
being seventy chains, and altitude thirty chains, at sixty- 
four dollars per acre. Ans. $6720, 

8. Required the area of an equilateral triangle whose 
side is twelve chains, and perpendicular ten chains. 

Ans. 6 acres. 

9. Required the area of a right- -angled triangle whose 
base is 140 rods, and perpendicular eighty rods. 

Ans. 35 acres.. 

10. Required the area of an isosceles triangle, whose 
base is forty chains, and altitude thirty-five chains, 

Ans. 70 acres. 

11. Required the area of a scalene triangle whose base 
is ninety-four rods, and perpendicular sixty reds. 

Ans. 17 acres, 2 roods, 20 rods. 

12. Eight gentlemen purchased a farm in the form of a 
right-angled triangle, the base thereof being 480 rods, and 
perpendicular 140 rods, and divided it equally. Required 
the share of each. Ans, 264 acres. 


To measure wood, bark and coal. 


Rule. Place the length, heighth and width on the right 
hand side of the line, and 128 on the left, or 8—4—4 on 
the left. — 


Nore. 128 cubic feet make a cord, and 2688 cubic 
inches make the bushel of coal or ais &c., and 2684 
cubje inches make the dry gallon. 


EXAMPLES.: 


1. How many cords of woodina load twelve feet long, 
eight feet high, and four feet wide ? . 
12— 3 
8 | 8— 
sei A 
—4 | 


| 3 cords, Ans. 


MENSURATION. 157 


2. How many cords of wood or bark in a load ten feet 
eight inches long, five feet four inches high, and four feet 
six inches wide? 


—3 | 32— 2 
—3 | 16— 
—2 | 9— 

—8 

—4 

—4 


| 2 cords, Ans. 


_ 3. How many bushels of charcoal in a load twelve feet 
long, eight feet high, and four feet wide, allowing 100 
bushels to the cord ? 


ie rae 
bi a ta 
ese) Oy. IRR 
—4 | 100 


es 


| 300 bushels, Ans. 


Norte. Since there are 2688 cubic inches in a heaped 
bushel, bearing the proportion to the cubic inches in a 
foot that fourteen does to nine, therefore, if the dimen- 
sions of your load be in feet, place the length, heighth, _ 
width, and nine on the right hand side of the line, and 
fourteen on the left hand side for bushels; but if you 
reduce all the dimensions to inches, you must divide 
by 2688, or the figures and numbers 8—16—7—3, which 
being multipled together give 2688. 


cs 


EXAMPLES. 


4. How many bushels of coal in a load ten feet eight 
inches long, five feet four inches high, four feet six inches 
wide ? 


—=3 | 32 
DISAAGL. Bx 4 
— ee Qs 

7 tt 1.9 


Fa 1152=164¢ bushels, Ans. 


158 MENSURATION. 


» The same reduced to inches. 
128 

cis pyle he 

—2 [6 |.54——..9 

"| 

ae 


oo 


7 | 1152—1644, Ans., as before. 


oe 


To find the solidity of a globe. 


Rule. Place the square of the diameter, the decimal, 
7854, 4 and t of the diameter of the globe, on the right 
hand side of the line, and the denomination next inferior 
to that corresponding to the answer on the left. 


EXAMPLES. 


1. Required the solidity of a globe, its diameter being 
thirty-six inches. 
36— 
36— 18 
»1804 
aa 
(; tae 


an a ae es 


| 14,1372 feet, Ans. 


2. Required the solidity of a globe, its diameter being 
four feet. 


48--- 4 - 
---12°} 48... 4 
---12 | ,7854-«- 2618 
SRA Bie Bg tO 
8 


ood 


| 33,5104 feet, Ans. 


8. Required the solidity of the planet Jupiter, its diame- 
ter being 89000 miles. Ans. 369121768400000 miles. 


MENSURATION,. 159 


‘a find the area of a Circle the Diameter and Curcumference 
being given. 


Rune. Place the diameter and circumference on the 
right hand side of the line, and 4, together with the next 
inferior number corresponding to the answer, on the left 


hand side. 


Proposition 1, Required the area of a circle whose 
diameter is 70 rods, and circumference 220 rods. 
—2 --4 |] 220— 110 
32 —160 | 70— 385— 7 
Ans. 24 acres 10 rods. 


2. Required the area of a circular whose diameter is 
350 rods and circumference 1100 rods. 

3. Required the area ofa circular field whose diameter 
is 140 rods and circumference 440 rods. 

4, Six gentlemen purchased a circular farm, the diame- 
ter of which was 210 rods and circumference 660 rods, 
and divided it equally between them. Required the share 
of each. Ans. 36 acres 0 roods 15 rods. 


To jind the area of a circle, the diameter being given. 


Rute. Place the square of the diameter and the decimal 
*7854 on the right hand side of the line, and the number 
next inferior to that corresponding to the answer, on the 


left hand side. 


Proposition 1. Required the area of a circle whose 
diameter is 80 rods. 
Solution. —2 —160 | 80— 
80— 40 
"1854 
Ans, 31 acres 1 rcod 25.56 rods. 


2. Required the area ofa circular field whose dimaeter 


is sixty-four rods. -) Ans. 20 acres 17 rods. 
3. How many acres in a circular field, the diameter 
being sixty rods 2 Ans. 17 acres 2 roods 27 rods. 


4, Required the area of a circular field, the diameter 
being one hundred and twenty rods. 
Ans. 70 acres 2 roods 29 rods, 


160 MENSURATION. 


5. Required the area of a circular farm, the radius (or 
semi-diameter) being eighty rods. 

Ans. 125 acres,,2 roods, 26 rods. 

6. Four gentlemen purchased a circular farm, the 

radius of which was forty rods, and paid equally. Re- 

quired the amount each paid, allowing they bought it at 

sixty dollars per acre. Ans. $471°24. 


To jind the superficial area of a globe, the circumference 
and diameter being given. 


Rute. Place the circumference and diameter on the 
right hand side of the line, and the next inferior number 
to that corresponding to the answer on the left hand side. 


Proposition 1. Required the superficial area of a globe, 
the diameter being 8 inches and circumference 24 inches. 
3 —6 —144 | 24— 
8— 4 11 feet Ans. 


2. Required the superficial area of a globe, the circum- 
ference being ninety-six inches and diameter thirty inches. 
Ans. 20 feet. 
3. Required the superficial area of a globe the diameter 
being 144 inches and circumference 452 inches. 
‘Ans, 452 feet. 
4, Required the superficial area of a ball, the diameter 
being thirty-six inches and circumference 112 inches. 
Ans. 28 feet. 
5, Supposing the earth’s diameter to be eight thousand 
miles and the circumference twenty-five thousand miles. 
How many square miles would there be on its whole sur- 
face ! Ans. 200000000. 
6. Required the number of square miles on the whole 
surface of Jupiter, the diameter being eighty-nine thou- 
sand miles and the circumference 280000 miles 4 
Ans. 24920000000, 


#: 


MENSURATION. 161 


To find the superficial area of a globe, the diameter being 
given. 


Ruut. Place the square of the diameter, the decimal 
°7854 and 4 on the right hand side of the line, and the 
number next inferior to that corresponding to the answer 
on the left hand side of the line. , 


Proposition 1. Required the superficial area of a globe, 
the diameter being thirty-six inches, 


36— 
—4 —144 | 36 
7854 
Le 


28.2744 square. 


2. Required the superficial area of a globe, the diame- 
ter being 144 inches. Ans. 452°3904 square feet. 
3. Required the superficial area of the earth, its diame- 
ter being eight thousand miles. 
Ans. 201062400 miles. 


Ee es - E 


To find the convex surface of a right cone. 


Rue. Place the circumference of the base and altitude 
on the right hand side of the line, and 2 and the next in- 
ferior number to that corresponding to the answer, on the 
left hand side. 


Prop. 1. Required the convex surface of a right cone, 
the circumference of whose base ig 72 inches, and slant 
height of altitude 24 feet. 


SO) Woe 
was $2) bea 
72 feet Ans. 


2. The circumference of the base of a right cone is 8 
feet, and slant height 20 feet, required its convex surface, 
Ans. 80 feet. 


162 MENSURATION. 


3. Required the convex surface of a right cone, the 
circumference of whose base is 96 inches, and slant height 
48 feet. Ans. 192 feet. 

4. The diameter of a right cone is 21 inches, and the 
slant height 36 feet; required the convex surface. 

: Ans. 98'9604 feet. 

5, Required the convex surface of a right cone, its 
diameter being 14 feet, and slant height 60 feet. 

Ans. 1319°472 feet. 

6. The diameter of a right cone is 45 feet, and the 
slant height 20 feet; required the convex surface. 

Ans. 141°372 feet. 

7. The circumference of the base is 10°75, and the 
slant height 18:25; what is the convex surface ? 

Ans. 98°09375. 


To find the convex surface of the frustrum of a right cone. 


Ruts. Place the sum of the perimeters of the two ends, 
and the slant height on the right hand side of the line, and 
2 and the number next inferior to that corresponding to 
the answer on the left hand side. 


Prop. 1. Required the convex surface of the frustrum of 
a right cone, the circumference of the greater end being 
30 feet, that of the lesser end 10 feet, and the length of the 
slant side 20 feet. 


40 
20— 10 


400 feet, Ans. 


2. Required the convex surface of the frustrum ofa 
right cone, the circumference of the greater end being 60 
foe! that of the lesser end 20 feet, and the length of the 
slant side 40 feet. Ans. 1600 feet. 

3. If a segment of twelve feet slant height be cut off a 
cone whose slant height is 60 feet, and’ circumference of 
“its base twenty feet. What is the surface of the frus- 
trum ? Ans. 576 feet. 


MENSURATION. . 163 


4, Required the convex surface of the frustrum of a 
right cone, the diameter of the greater end being 22 feet, 
that of the lesser end seven feet, and the length of the 
slant side 16 feet. Ans. 728,8512 feet. 


eee ee 


To find the solidity of a cone or pyramid. 


Rute. Place the square of the diameter, the decimal 
*7854, and altitude on the right hand side, and 3 and the 
number next inferior to that corresponding to the answer 
on the left hand side. 


Prop. 1. Required the solidity of a cone, the diameter 
being twenty inches-and altitude or perpendicular height 
twenty-four feet. 

—6 —144]20 | 
32/20 
.7854— 1309 
| a4 


rs 


17,4533 feet. 


2. Required the solidity of a conical church spire, the 
diameter being twelve feet, and perpendicular height 
sixty feet. Ans, 2261,952 cubic feet. 

3. The diameter of a cone is twenty feet and its per- 
pendicular height twenty-four feet. Required its solidity. 

: Ans. 2513,28 feet. 

4. Required the solidity of a conical block of marble, 

its diameter being 9 feet, and altitude twenty-four feet. 
| Ans. 508,9392 cubic feet. 

5. Required the value of a conical marble monument 
of $12 50 per foot, the diameter of whose base is twelve 
feet, and perpendicular height thirty-six feet. 

Ans. $16964,64 


o 


164 » MECHANICAL OPERATIONS. 


MECHANICAL OPERATIONS. 


The two arms of a lever and the power being given, to find 
what weight that power will equiponderate. 


Rule. Place the length of the arm to which the power 
is applied, and the power on the right hand side of the 
line, and the length of the other arm on the left hand 
side. 


Prop. 1. There is a lever thirty feet long divided by the 
fulcrum into two arms, one of which is twenty feet, the 
other ten feet in length. Required the equiponderating 
weight on the short arm when 120 pounds is suspended 
at the extremity of the long arm. | 


A 2053 
; 120 


| 240 pounds, Ans. 


2. The arms of a lever are, the one thirty feet and the 
other four feet inlength. What weight will a power of 
160 pounds at the extremity of the long arm balance at 
the extremity of the short arm? Ans. 1200 Ibs. 

3. How many lbs. will a power of nine lbs. placed fif- 
teen feet from the fulcrum of a lever, support at the ex- 
tremity of the other arm two feet in length? Ans. 674. 


The arms of a lever and the weight being given, to find the 
power. 


Ruut. Place the weight and the length of the arm to 
which it is suspended on the right hand side of the line, 
and the length of the other arm on the left hand side. 

+ 


Prop. 1. A weight of twenty tons is suspended to an 
_arm of a lever six inches in length. What weight at the 
extremity of the other arm forty feet in length will balance 
the same ? \ 

[Solution, next page. | 


MECHANICAL OPERATIONS. — 165 


--2 ---12 | 20--- 10--- 5 ewt. Ans. 
—2 —40 | 20--- 
| Satis | 

2. A weight of fourteen hundred pounds is suspended 
to an arm of a lever eight feet in length. What weight 
at the extremity of the other arm fourteen feet in length: 
will balance the same? | Ans. 800 lbs. 
3. A weight of four hundred tons is suspended to an 
arm of a lever ten inches in length. Required the weight 
at the extremity of the other arm five feet in length that 
will balance the same. Ans. 662 tons. 
4, A weight of 7200 lbs. is suspended to an-arm of a 
lever three feet in length. What weight at the extremity 
of the other arm nine feet in length will balance the same ? 
Ans. 1;); tons. 


The diameter of the wheel, the diameter of the axle and the 
power being given, to find the weight. 


Rule. Place the diameter of the wheel and the power 
applied on the right hand side of the line, and the diame- 
ter of the axle on the left hand side. 

Prop, 1. If the diameter of the axle be six inches and 
that of the wheel six feet, what weight attached to the 
axle will sixteen lbs., attached to the wheel, balance? 

2 ress 
—6 1,12 
16 


[+192 Ibs. Ans. 

2. Ifthe diameter of the axle be eight inches and that 
of the wheel twenty-four feet, what weight attached to the 
axle will 144 lbs. attached to the wheel balance ? 

: Ans. 242 tons. 

3. If the diameter of the wheel be thirty-six feet and 
that of the axle four inches, what weight attached to the 
axle will twelve cwt. attached to the wheel balance ? 

. Ans. 64£ tons. 
4, Supposing the diameter of the wheel to be forty- 
eight feet and that of the axle ten inches, what weight 
attached to the axle would one hundred tons attached to 
the wheel balance ? Ans. 5760 tons. 


166 MENSURATION. 


5. Supposing the diameter of the wheel to be sixty feet 
and that of the axle twelve inches, what weight attached 
to the axle would four hundred tons attached to the wheel 
balance ? Ans. 2000 tons. 


The diameter of the wheel, the diameter of the axle, and the 
weight being given, to find the power. 


Rule. Place the diameter of the axle and the weight on 
the right hand side of the line, and the diameter of the 
wheel on the left hand side. 


Prop. 1. If the diameter of the axle be six inches and _ 
the diameter of the wheel twelve feet, what power will 
balance a weight of 360 lbs. ? 


weeeshe Bae 
—12 | 360— 30— 15 lbs. Ans. 


2. Ifthe diameter of the axle be eight inches and that 
of the wheel sixteen feet, what power will balance a weight 
of 2880 lbs. ? Ans.-120 Ibs. 

3. If the diameter of the wheel be twenty.four feet and 
that of the axle four inches, what power will balance a 
weight of forty tons ? ; Ans. 114 ewt. 

4, If the diameter of the axle be nine inches and that 
ofthe wheel eight feet, what power will balance a weight 
of 144 tons ? Ans. 13} tons, 


The length, height of the plane and power being given, to 
determine the weight. 


Rule. Place the power and the length of the plane on 
the right hand side of the line, and the perpendicular 
height on the left hand side. 


Prop. 1. If the length of an inclined plane be sixteen 
feet and the perpendicular height four feet, what will a 
power of thirty-two pounds sustain ? 

[Solution, next page. | 


MENSURATION. 167 


09 


"128 nade Ans. 


2, What weight will four tons sustain on an inclined 
plane one hundred and forty-four feet in length, and per- 
pendicular height four feet ? Ans. 144 tons. 

3. If the length of an inclined plane be eighty-four 
feet and perpendicular height three feet, what weight will 
a power of twenty tons sustain? Ans. 560 tons. 
_ 4, What weight will twelve tons sustain on an inclined 
plane ninety- SIX tech) in length and perpendicular height 
six feet? Ans. 192 tons. 

5. If the Pine. of an inclined plane be seventy-two 
feet and perpendicular height 32 feet, what weight will a 
power of one hundred sixty cwt, sustain? Ans. 1724 tons. 


The length, height of the plane, and weight being given, to 
Jind the power. 


Rule. Place the weight and height of the plane on the 
right hand side of the line, and the length on the left 
hand side. 


Proposition 1. What power will balance one hundred 
twenty-eight pounds on an inclined plane, the length of 
which is sixteen feet, and perpendicular height four feet ? 


---16 | 128— 8 
4 


32 pounds, Ans. 


2. What power will balance twelve tons on an inclined 
plane, the length of which is twenty-four feet, and per- 
pendictilar height six feet ? Ans. 3 tons. 

3. What power will balance twenty tons on an inclined 
plane, the length of which is seventy-two feet and per- 
pendicular height four feet ? Ans. 1} tons. 


168 MENSURATION. 


4. What power will balance one hundred and forty-four 
cwt. on an inclined plane, the length of which is ninety- 
six feet, and the perpendicular height nine feet ? 

Ans. 133 cwt. 

5. What power will balance thirty-six tons on an in- 
clined plane, the length of which is two hundred and forty 
feet and perpendicular height eight feet?. Ans. 24 cwt. 


The thickness of the head, the length of the side and the power 
acting upon the head of the wedge being given, to deter- 
mine the force produced on the side. 


- Rule, Place the length of the wedge and the power on 
the right hand side of the line, and the thickness of the 
head on the left hand side, 


Proposition 1. If the length of a wedge be 12 inches, 
the thickness of the head 3 inches, and the force applied 
64 pounds, what will be the resistance at the side ? 


—3 | 12— 4 
64 


256 pounds, Ans. 


2. If the length of a wedge be twenty inches, the thick- 
ness of the head four inches, and the force applied one 
hundred and forty-four pounds, what will be the resistance 
at the side ? Ans. 62 cwt 

3. If the length of a wedge be thirty-six inches, the 
thickness of the head six inches, and the force applied 
nine paaeee sixty pounds, what will be the resistance at 
the side ? Ans. 2+ tons. 

4, If the length of a wedge be forty-eight inches, the 
thickness of fhe, head eight inches, and the force applied 
ten thousand eight hundred pounds, what will be the 
resistance at the side ? Ans. 284? tons. 

5. Ifthe length of a wedge be sixty inches, the thick- 
ness of the head five inches, and. the force applied eight 
tons, what will be the resistance at the side? . 

Ans, 96 tons. 


MENSURATION. 169 


The length of the side, the thickness of the head, and the 
resistance upon the fide of a WEDGE pete given, to. find 
the force acting upon the head, 


Rue. Place the resistance at the side and the thick- 
ness of the head on the right hand side of the line, and the 
length of the side of the wedge on the left hand side. 


Proposition 1. If the resistance at the side of a wedge 
be twenty thousand pounds, the length of the wedge 
twenty inches, and the thickness of the head three inches, 
what force is required to be applied to counteract the 
resistance at the sides ? 

—20 | 20— 000 
| 3 


eres 


3000 pounds, Ans. 


2. If the length of the wedge be thirty-two inches, the 
thickness of the head two inches, and the resistance at 
the side be twenty: -five thousand six hundred pounds, 
what must be the force upon the head, no alluwance 
being made for friction ? Ans. 1600 lbs. 

3. Ifthe resistance at the side of the wedge be twelve 
tons, the length of the wedge twenty-four inches, and the 
cep ee of ‘the head four qncies, what force is required 
to be applied to counteract the resistance at the sides 2 

Ans. 2 tons. 

4. If the length of the ie be forty eight inches, the 
thickness of the head six ES and the vardatn ace at the 
side twenty-four tons, what must be the force upon the 
head ? = at DS. 3 LONES 


The distance between the threads of a Screw, the length of 
the lever, and power applied being given, to Jind the 
weight. 


Rule. Place the circumference of the circle described 
by one revolution of the lever and the power applied on 
the right hand side of the line, and the distance between 
the threads of the screw on the left hand side. 

16 


170 . MENSURATION. 


Proposition 1. Ifthe threads of a screw be two inches 
asunder, the lever thirty-five inches in length, and a 
power of sixty pounds be applied to the end of the lever 
what weight will be required to produce an equilibrium ? 


—2 | 220. 
60— 30 


6600 pounds, Ans. 


2. If the threads of a screw be three inches apart, the 
Jever 244 inches in length, and a power eighty pounds ‘be 
applied to the end of the lever, what weight will be 
required to produce an equilibrium 1 Ans. 41062 Ibs, 

3. Should the threads of a screw be 24 inches asunder, 
the lever twenty-eight inches in length, and a power of 
four tons be applied to the end of the lever, what weight 
will be required to produce an equilibrium ? 

, Ans, 2813 tons. 

4. Should the threads of a screw be 4,4, inches asun- 
der, the lever thirteen inches in Jength, and a power of 
four tons be applied to the end of the lever, what weight 
will be required to produce an equilibrium ? Ans. 47 cwt. 


The weight of the lever and the distance between the threads 
of a screw being given, to find the power requisite to pro- 
duce an equilibrium. 


Rule. Place the given weight and distance between the 
threads of the screw on the right hand side of the line, and 
the circumference of the circle, described by one revolu- 
tion of the lever, on the left hand side. 


Proposition 1. How many pounds applied to the end 
of a lever 36,4; inches in length will balance twenty tons 
upon a screw whose threads are two inches asunder? 


| 20 pounds, Ans. 
—2 —224 | 112— 


* 


MENSURATION; 171 


2. How many pounds applied to the end of a lever 
thirty-five inches in length will balance fifteen tons npome 
screw whose threads are 25-inches apart? Ans. 3.4; cwt. 

3. Required the number of pounds requisite to pr roduce 
an equilibrium of twenty-four tons upon a'screw whose 
threads are 22 inches asunder, and: lever twenty-eight 
inches. Ans. 65% cwt. 

4. The threads of a screw are two inches asunder, the 
length of the lever seventy inches, required the number 
of pounds ‘to produce an equilibrium, the weight peed 
to the lever being seventy-two tons. Ans. 6,5 tons. 


ee 


To ascertain the atmospheric pressure upon a cylinder. 


Rule. Place the square of the number of inches in the 
diameter, and 165 on the right hand side of the perpen- 
dicular ae, and 14 on the left hand side. 


Prop. 1. Required the atmospheric pressure upon a 
cylinder whose diameter is twenty-eight inches. 


31d) p28dw 2x. 
eB |-28--— 
2 ---4 1 165— 824 cwt. Ans. 


2. Required the atmospheric pressure upon a piston of 
a steam engine whose diameter is fifty-six inches. 
Ans. 16 tons 10 cwt. 
3. The diameter of a cylinder is 2} inches. Required 
the atmospheric pressure. Ans. 643 Ibs. 
4. Required the atmospheric pressure upon a piston of 
a commun pump, the diameter of which is seven: inches. 
Ans. 5774 Ibs. 
5. Required the atmospheric pressure upon a piston of 
a 2 erin vessel whose diameter is seventy inches. 
Ans. 25 tors 15 ewt. 2 qrs. 14 lbs. 
6. The diameter of a cylindrical vessel is 14 inches. Re- 
‘quired the atmospheric pressure upon its piston. 203 cwt, 
7. The diameter of a cylindrical vessel is thirty-eight 
inches. Required the atmospheric pressure on its piston. 
Ans. 7 tons 11 cwt. 3 qrs, 22 Ibs. 


172 MENSURATION. 


+ Wie the number We bails contained: in a finished triangu- 
«lar pile.» 


sedis, Place the number of balls contained in the bot- 
tom row increased by 2,:'the number of balls contained in 
the bottom row increased by. 1, and the number of balls 
contained.in the bottom row on the right hand side of the 
perpendicular line, and 6,on the eft hand side. 


Prop. 1. Required the number of balls contained ina 
finished triangular pile, the bottom row consisting of 
eight on a side. 


: 10 
--+2 ---6 | 9--- 3 
18---"4 
120 Ans. 


2. Required the number of balls contained in a finished 
triangular pile, the bottom row consisting of thirty on a 
side. Ans. 4960. 

3. How many balls are contained in a finished triangu- 
lar pile, each side of whose base contains twenty balls ? 

Ans. 1540. 

4. Required the number of balls contained in a finished 
triangular pile, the bottom row consisting of sixty on a 
side. Ans. 9920, 


To find the number of balls contained in a finished square pile. 


Rute, Place twice the number of balls contained in the 
side of the square increased by 1, the number of balls 
contained in the side of the square increased by 1, and the 
number of balls contained in the side of the square on the 
right hand side of the perpendicular line, and 6 on the left 
-hand side. 

Prop. 1. Required the number of balls contained in a 
finished square pile containing twelve in each side. 


[Solution, next page.] 


MENSURATION. : 173. 


ipoa x 
---6 1-13 
[ipie'2 
650. Ans. : 


2. Required the number of balls contained in a finished 
square pile, the lower tier containing thirty in rot ae 
“Ans. 9455. 
3. Required the number of balls contained in a finished 
square pile, each side containing twenty balls. 
Ans. 2870. 
4, Required the unber of balls contained in a finished 
square pile, each side containing twenty-three balls. 


To ascertain the number of balls contained in a finished rec- 
. langular Piero 


Rule. To twice the number of courses rercared by 1, 
add the produet of the number less by 1 in the top row 
multiplied by 3, and place the sum, together with the 
number of courses increased by 1, and the number of 
courses on the right hand side of the line, and 6 on the 
left hand side. 


Proposition 1..The number of courses in a. finished 
rectangular pile is thirty, and the number in the top row 
is thirty-one, required. the number contained in the-pile. 


| 151 
beeen 1 ae 
_180,-+ 5 


23405 balls, Ans. 


2. The number of courses in a finished rectangular pile 
is twenty, and the number in the upper course is twenty- 
four, required the number contained 1 in said pile. 

Ans. 7,700. 
3, The number of shot in the upper course of a finished 


_ 


174 MACHINERY. 


rectangular pile is forty-one, and the number of courses, 
thirty, how many shot are contained in said pile, 

Ans. 28055. 

4, How many shot in a-finished rectangular pile, the 

length of the bottom course being fifty-nine and its breadth 

twenty ¢ Ans. 11060. 


MACHINERY. 


To ascertain the number of revolutions that a drum, pulley or 
spindle will make, when connected together by belts or bunds, 
the velocity of one and diameter being’ given. 


Rue. Place the velocity and diameter of the drivers 
on the right hand side of the perpendicular line, and the 
diameter of the driven on the left hand side of the line. 


Prop. 1. A belt connects a drum of two feet in diame- 
ter, making forty revolutions in a minute with one of four 
inches in diameter. Required the velocity of the smaller 
drum. fi pe ia 2 | 
—4 | 40 


240 revolutions per min. Ans. 


2. How many revolutions will a spindle of two inches 
in diameter make, connected to a drum of three feet in 
diameter performing thirty revolutions per minute ? 

Ans. 540. 

3. A drum of four feet in diameter performs sixty revo- 
lutions per minute. Required the diameter of that drum, 
whose velocity is 576 revolutions per minute. 

Ans. 5 inches. 

4, What is the twist of yarn per inch, spun on a mule 
with the following geer, pullies, &c.: geer on front roller, 
54 teeth ; geer on lower end of tumbling shaft, 27 teeth ; 
geer on upper end of tumbling shaft 44 teeth; geer 
on fly wheel shaft, 50 teeth; diameter of fly wheel, 36 
inches; diameter of twist pully, where rim band runs, 
16 inches; diameter of twist pully, where drum bands 


SQUARE ROOT. 175 


run, 124 inches; diameter of drum, where the drum band 
runs, 10$ inches: diameter of drum where spindle band 
runs, 10 inches ; diameter of spindle whirl, 1 inch ; diame- 
ter of front roller, 1 inch. 


Solution. —27 | 54—- 2— 
—2 —50 | 44— 2— 
—2 —8 —16 | 36— 12— 3 


—2 | 25— 

—3 —21 | 2— 
10— 5 

—22 | 7— 


ee, 


15 turns per inch Ans. 


5, What is the draft of a spinning frame, front roller 
14 inches; diameter of back roller 8 of an inch; pinion 
on front roller 40 teeth, stud 84 to 21 teeth ; geer on back 
roller 50 teeth. | 


eh Ga 
6 158... 

—5 —40 | 84— 4— 
—21 | 50— 10 


10 is to 1, Ans. 


SQUARE ROOT. 


The square of a number is the product arising from a 
number multiplied into itself. 

The extraction of the square root isthe finding of such 
a number as, being multiplied itself, will produce the 
number proposed. 


Rute. Separate the given number into periods of two 
figures, each beginning at the units’ place. 

Subtract from the first period at the left hand the great- 
est square it contains, setting the root of that square as a 
quotient figure, and doubling said root for a divisor, and 


bring down the second period to the remainder for a divi- 
dend. 


176 ‘CUBE ROOT: 


Try how often the said divisor (with the figure used in 
the trial thereto annexed) is contained in the dividend, and 
set this figure in both the divisor and root; then multiply 
and subtract, as in division, and bring down the next 
period. Double the ascertained root for a new divisor, 
and repeat the process to the end. 


Proor. Square the root, adding in the remainder, if 
any, and the result will equal the given number. 
EXAMPLE. 
What is the square root of 30138,696025? 


*x * * 
1.) 30138, 696026 (173,605 
le aed 


27) 201 
7 189 


343) 1238 
3 1029 
3466) 20969 
6 20796 
347205) 1736025 
1736025 


ew 


‘CUBE ROOT. 


The cube of a number is the product of that number 
multiplied by its square. 

The extraction of the cube root is the finding of such a 
number as, being multiplied into its square, will produce 
the number proposed. 


Rue. Point off the given numbers into periods of three 
figures each, and find the nearest cube to the first period ; 


CUBE ROOT. 177 


subtract it therefrom, and put the root) in the. quotient ; 
then thrice the square of this root will be the trial divisor 
for finding-the next figure. es 

Set off a little to the left. the next figure; with thrice 
the peceding figure of the root; multiply this by the last 
figure, and set this under the trial divisor, remove it two 
figures to the right, and the sum will be the true divisor. 

Under this divisor put the square of the last period 
figure of the root, which add to the two sums above, and 
the sum will be the trial divisor for finding the next figure 
of the root; then the true divisor is found, as before. 


EXAMPLES. 
1. What is the cube root of 205379 2? 
3 multiplied by 275 205379(59 
$595 & 1481 a LOB 
8931 80379 


80379 


2. What is the cube root of 122615327232 2 


3x2 = 48 122615327232(4968 Ans. 
129-1161 64 
5961 58615 
81 53649 
7203 4966327 
1476 8856 4374936 
729156 591391232 
36 591391232 
738048 


17888 119104 


73923904 


17? 


178 CUBE ROOT. 


3. Require to extract the cube root of 2205 to 19 places. 


507 2205(13,01575997906296270 
S001 ... .- ,3901 2197 
507,3901 da 8 | 
0001 5073901 
507,7803 2926099 
39,085 .. . 195175 2539877375 
507,975475 386221625 
25 355738604893 
508, 17075 30483020107 
39,0457 . . . . 2733199 25411364591375 
508, 19800699 5071655515625 
A9 4574066360515 
508, 22533947 497589155110 
39,04715 . .... 19523575 457406983963 


508, 2272918275 

25 

508, 2292441875 
39,0,4,7,4,5,9 3514253 


re eee 


508,229595612,8 


508,229947038 
35143 


508,22998218,1 
508,23001732 
273 


-~ 


ee 


508,2300200,5 


508,2300228 
4 


eee eee ee 


5,0,8,2,3,0,0,2,3,2 


a Snare 


en ee 


40182171147 
309976101403 


es Se 


4606069744 
4574070209 


31999535 
30493801 
1505734 
1016460 
489274 
457407 


31867 | 
30494 


1373 
1016 


357 
355 


OBE ROOT. 179 


APPLICATION OF THE SQUARE AND CUBE ROOT. 


1. A certaln pavement is made exactly square, and each 
side of it contains 97 feet, how many square feet are con 
tained therein ? Ans. 9409. 

2. A certain square pavement containing 20736 square 
stones, all of the same size, what number is contained in 
one of its sides ? Ans. 144, 

3. A certain number of men gave $3,61 for a charitable 
purpose, each man gave as many cents as there were men, 


how many men were there ? Ans. 19. 
4. If 484 trees be planted in asquare orchard, how many 
must there be in a row each way ? Ans, 223 


Norr. The square of the longest side of a right angle 
triangle is equal to the sum of the squares of the other two 
sides, and consequently the difference of the square of the 
lungest and either of the others is the square of the re- 
maining one. 

5. The wall of a certain fortress is 17 feet high, which is 
surrounded by a ditch 20 feet in breadth, how long must 
a ladder be to reach from the outside of ‘the ditch to the 
top of the wall ? Ans. 264 feet. 

6. A line of 36 yards long will exactly reach from the top 
of a fort to the opposite bank of a. river, known to be 24 
yards broad, the height of the wall is requir ed? 

Ans. 26,83 yds. 

7. Suppose a ladder60 feet long be so planted asto reach 
a window 37 feet from the ground on one side of the 
street, and without moving it at the foot, will reach a win- 
dow 23 feet high on the other side, what was the breadth 
of the street ? Ans. 102,64 feet. 

8. A certain tree is broken off 8 feet from the ground, and 
resting on the stump touches the ground at the distance 
of 12 feet, what is the length of the part broken off ? 

Ans. 14,42 feet. 

9. Two ships sail from the same port, one due east and 

the other due north, what is the distance between them 


when one has sailed 100 miles‘and the other 168 miles ? 
Ans. 1954-F miles, 


180 CUBE ROOT. 


10. A man shot a bird sitting on the top of a steeple S0 
feet high while standing at the distance of 60 feet from its 
base, how far did he shoot ? -Ans. 100 feet. 

11. Two boys were playing with a kite, the line of which 
was 520 feet in length, when the string’was all out, one 
of them standing directly under it and the other holding 
the string, the distance between them was 312 feet, what 
was the perpendicular height ofthe kite, Ans. 416 feet. 

12. If a pipe whose diameter is 14 inches fill a cistern 
in 5 hours, in what time will a pipe of 34 inches diameter 
fill the same 4 Ans. 543 minutes. 

13. Suppose a cellar to be dug that shall be 12 feet 
every way, in length, breadth and depth, how many solid 
feet of earth must be taken out to complete the same ? 

3 Ans. 1728. 

14, A gentleman laid out €691 4s.in cloths, but forgot 
the number of pieces purchased, also how many yards 
were in each piece, and what they cost him a yard, but 
he remembers that they cost him as many shillings a yard 
as there were yards in each piece, and that there was just 
as many pieces. Query, the number purchased? 

Ans. 24. 

15. What is the side of a-cubical mound equal to one, 
144 feet long, 108 feet broad and 24 feet deep? 

| Ans. 72 feet. 

16. Ifa ball 6 inches diameter weigh 8 lbs. what is the 
weight of another 12 inches diameter 1 Ans. 64 Ibs. 

17. What would be the value of a globe of silver one 
foot in diameter, if a globe of the same, one inch diameter, 
be worth $6? | Ans. $10368. 

18. Suppose the diameter of the sun to be 110 times as 
large as that of the earth, how many bodies like the earth 
would be required to make one as large as the sun ? 

: Ans. 1331000. 

19. If a globe of silver one inch in diameter be worth 
$6, what is the diameter of another globe of the same 
metal, worth $10368? _ 


ee a 


Lo extract the square root of a vulgar fraction. 


Rue. Reduce the fraction to its lowest terms, then ex- 
tract the square root of the numerator for a new numera- 


PROBLEMS. 181 


tor, and the square root of the denominator for a new de- 
nominator. - 7 3 

If the fraction of a surd (i. e.) a number whose root can 
never be exactly found, reduce it to a decimal and ex- 
tract the root from it. 


PROBLEMS. 


PROBLEM I. 


When the sum of two numbers is given with their differ- 
ence, to find those numbers. 


Rue. To half the sum add half the difference, and their 
sum will be the larger number. From half the given 
sum, take half the difference, and the remainder will be 
the smaller number. 


EXAMPLE. 


The sum of two numbers is 98, and their difference 14; 
what are those numbers ? 


2)98 2)14 
49 half the sum. 7 half their difference. 
vise 49 x 
- 42 smaller number. 56 larger number. 


PROBLEM II. 


When the sum of two numbers is given, and the differ- 
_ ence of their squares to find those numbers. 


Rute. Divide the difference of their squares by the sum 
of their numbers, and the quotient will be the difference of 
the numbers; then by Problem 1st find those numbers. 


182 PROBLEMS, 


EXAMPLE. 


The sum of 2 numbers is 60, and the difference of their 
squares 1200—what are those numbers 4? 


60)1200 
20 diff. of the num, By Problem Ist. 
2)60 | 2)20 
30 | 10 half, 
10 30 
20 40 greater num. 


There is a pole 100 feet high ; how far from the ground 
must the pole be cut off, and resting on the. stump the 
end shall reach .the ground 60 feet from the bottom of the 
stump ¢ Ans. 32 feet. 


Nore. In this question the length of the pole is the sum 
of two numbers, and the square of 60 is the difference of 
their squares. 


PROBLEM III. 


When the sum of two numbers is given, with the sum 
of their squares to jind those numbers. 


Rue. From half the sum of the squares subtract the 
square of half the sum of the numbers, and the square 
root of the remainder will be half the difference of those 
numbers; then by Problem Ist find those numbers. 


EXAMPLE. 


The sum of two numbers is 60, and the sum of the. 
squares 2000—what are those numbers ? 
2)2000 sum of the squares, 


1000 half the sum of squares, 
960 square of 4 the sum of the numbers. 
100 square root of remainder—10 the difference 
of the numbers required. 
2)60 sum of the numbers, 


30 — 30—half the sum of the number, 
10sub’t.10 add 


— 


20 and 40—numbers required. 


_ PROBLEMS. 183 


There is a right angle triangle; the hypotenuse is ten 
feet, the base and perpendicular 14 feet—the length of the 
base and perpendicular each required. Ans. 6 and 8. 


PROBLEM IV. 


When the sum of the squares of two numbers is given, 
and the difference of the numbers to find those numbers. 


Rute. From the sum of the squares subtract half the 
square of the difference; the square root of half the re- 
mainder will be half the sum of those numbers; ; then by _ 
Problem 1st find those numbers. 


EXAMPLE. 


The sum of the squares of two numbers is 2000, and the 
difference of the numbers 20. What are those numbers? ? 
Sum of the square, 2000 
Half the square of the difference, 200 


2)1800 


Half the number, 900 
30 square root of half the remainder, which is half the 
sum of the numbers; to which, if 10 be added, the sum 
will be 40; if subtracted it will be 20. 

A and B played at hazard, B losing would play no lon- 
ger, and on counting they found the difference of their 
sums to be 12, and the sum of the squares 272 ; how many 
dollars had each when they quit? Ans. A 16, B. 4, 

There is a right angled triangle whose hypotenuse is 10 © 
feet; the difference of the base and perpendicular 2 feet,— 


the length of base and perpendicular, each, required. 
_Ans, 6 and 8. 


PROBLEM Y. 


The product of two numbers given, and the sum of the 
numbers to find the numbers. 


184 PROBLEMS. 


Ruiz. From the square of half the numbers subtract 
the product; the square root of the remainder will be 
half the difference of the numbers. Numbers then found 
by Problem ist. 


EXAMPLE. 


The product of two numbers is 20; the sum of the num- 
bers is 12. What are the numbers ? 


Product 20—4)12 
- 6 half the sum, 
6 6 
— 4 
36 — 
20 10 


. Remainder 16 square root 4, which is $ the 
difference of the numbers. 
There is a right angled triangle whose area is 24 feet, 
base and perpendicular 14 feet,—the length of base and 
perpendicular required. Ans. 6 and 8. 


* PROBLEM VI. 


The product of two numbers and the difference of the 
numbers given to find the numbers. 


Rute. To the product add the square of half the differ- 
ence; the square root of the sum will be the sum of’ half 
the numbers. ‘Then proceed as Problem 4th. 


EXAMPLE, 


The product of two numbers is 32, and the difference 
of the numbers 4. What are the numbers? 32 product, 
Ans. 8 and 4. 


[Solution on next page.] 


PROBLEMS. | 185 


2)4 difference, 
2 half the difference, 
9 } 


32 product, 4 square of 4 the difference. 
4 


36(6 square root, half the sum of numbers, 
6 


36 square of half the numbers, 
32 product, 
4(2 square root, 

6 6 sum of half the numbers, 

me 

8&4 the numbers required. 

A right angled triangle, the areais 24, the difference of 
the base and _ perpendicular is 2 ,—required thé length of 
the base and perpendicular each. © Ans. 6 and 8. 

A merchant mixed thirty dollars’ worth of American 
gin with thirty dollars’ worth of Holland gin. There were 
60 gallons of the mixture; the Holland gin was worth 6 
shillings per gallon more than the American. How many 
gallons were there of each, and what per gallon? 

Ans. 40 galls. at 61s. and 20 galls. at 12s. 


Note. 1st find the medium price. 

2d. Suppose any two numbers whose difference is 
given, one of which must be more than the given price. 

3d. By allegation alternate, find a number that is worth 
the medium price. 

4th. Find the value of the mixture, and if the two 
quantities which compose it amount to equal sums, the 
supposed prices are right... If their sums are not equal, 
add them together and halve the sum. 

5th. By Problem 6th find two numbers whose differ 
ence is the difference of the least supposed number and the 
quantity that may be placed against it. By the process 


186 PROBLEMS. 


- in allegation alternate and their products the half value of 
the mixture, one of the numbers so found will be the re- 
quired price, and the given difference being added to it 
will be the other required number. Then find the quan- 
lity of each by the single rule of three. 
Suppose 10 and 4 medium 8 
Sea oe 5,.0 
Age 20 8 Vise k 


A&6 


6 the American gin 40 gallons, 


2)48 


24 


1 6 
25(5 | 12 Holland gin 30 gallons. 
PROBLEM VII. 


The sum of the square of two numbers given, and the 
product to fipd those numbers, 


Rute. rom the sum of the square take twice the pro- 
duct, and the square root of the remainder will be the dif- 
ference of the numbers. Then proceed as in Problems 
4th and Ist. 

Or, to the sum of the squares add twice the product; 
the square root of the sum will beé'the sum of the numbers. 
Then find the numbers by Problem 3d. 


EXAMPLE. 


The product of two numbers is 8, the sum of their sepa- 
rate squares is 20. What are the numbers ? 
Ans. 2 and 4. 
20 sum of the squares, 
16 twice the product, 
36(6 square root and sum of the numbers. _ 
10 half the sum of the numbers, 
9 square of half the numbers. 


3-3 1(1 square root of the difference and 4 the dif- 
bssok ference of the numbers. 


4 & 2 the numbers required. 


PROBLEMS, . 82 


The hypotenuse of a right angled triangle is 10 feet, the 
area is 24 feet,—what is the length of the base and per- 
pendicular, each ? Ans. 6 and 8. 


Note. Twice the area is the product, and the square of 
the hypotenuse is the sum of the squares. 


PROBLEM Vilt: 


The sum of the product of two numbers with the square 
of one of them given ; also the sum of the numbers given 
to find those numbers. 


Rule. Divide the sum of the product and square by the 
sum of the numbers; the quotient is the number to be 
squared. 

The product of two numbers with the square of one of 
them is 12, and the sum of the numbers is 6,—what are 
the numbers ? Ans. 2 and 4. 

Sum of the numbers 6 | 12 the number given, 

Pg ae bo 
— | 2 smaller number. 
Large number 4 


PROBLEM Ix. 


The relation of two numbers and their products given 
to find their numbers. 

Rule. Divide the product by the product of the num- 
bers denoting the relation. The square root of the quo- 
tient multiplied by each of the figures denoting the rela- 
tion; the product will be the numbers required. 


EXAMPLE. 


The relation of two numbers is as 3 to 4, the product 
48 ; numbers required. 


4 12)48 
3 sia 
ee 4 
12 i 
oye tot 
2 2 


6 & 8 answer. 


188 PROBLEMS, 


The area of a right angled triangle is 24 feet, the base 
is to the perpendicular as 3 to 4—what is the length. of 
each ? Ans. base 6, perplr. 8. 


PROBLEM X. 


The relation of two numbers given and the sum of their 
squares to find the numbers, 


Rule: Divide the sum ofthe squares by the sum of the 
squares denoting the relation. The square root of the 
quotient multiplied by each of the numbers denoting the 
relation of the product will be the numbers required. 


EXAMPLE. 


The relation of two numbers is as 3 to 4, the sum of 
their squares is 100—what are the numbers ? 


3.4 ore 25)100 
“22 3 A —--: 
— ~ | -— 4(2 sqr. root. 
Ans. 6&8 9 16 
9 


25 sum of the squares. 
The hypotenuse of a right angled triangle is 10 feet, 
the base to the perpendicular is as 3 to 4—what is the 
length of each ? Ans. base 6, perplr. 8. 


PROBLEM XI. 


Having the area and the sum of the sides of a right an- 
gled triangle given to find the sides. 


Rule. Divide the square of double the area by the 
square of the sum of the sides; the square root of the 
quotient subtracted from half the sum of the sides, the re- 
mainder will be the longest side or hypotenuse, which, 
being subtracted from the sum of the sides, the remainder 
will be the sum of the remaining sides, which find by 
Problem 3d. 


Nore. The square of the hypotenuse will be the sum 
of the squares of the two remaining sides. 


PROBLEMS, 189 


' EXAMPLE. 


Let the area of a right angled triangle be 6 rods, and 
the sum of the sides 12 rods,—what is the fongth of each 
ade | t | 
2 

12 double the area, 
,12 e 
144 square of double. the area, 
6 half the sum, | 12 square of sum of the sides, 
1 of sides, 12 
5 hypotenuse, | 144(144 
12 1(1 square root of quotient, 
5 5 hypotenuse, 
ee 5 


7 sum of sides. — ; ; 
2)25 square of sum of sides, 
12,5 half the sum of the sqrs. 
2(7 : 
. 3,5 half the sum of the numbers, 
“330 
175 
105 
12,25 squares of half the sum of the numbers, 
12,50 half thesum of the squares, 
12,26 
,25(,5 square root, 
_ 2)7 sum of the sides, 
3,5 = -.3,5 
9 9 


3, and 4, Ans, 


190 PROBLEMS. 


PROBLEM XII. 


When a sum is involved in its square, and the sum is 
required. i | 

‘Rule. Proceed as in the extraction of the square root, 
but deduct from the given sum the quotient figures, units 
from units, tens from tens, &c., and in decimals, tenths 
from tenths, hundredths from hundredths, &c., and the 
quotient will be the involved root required. 


EXAMPLE. 
15363,3525 is a number with its square root involved,— 
what is the number 2 
15363,3525)12345 
1 


22(58 
44 


243(963 
729 
234 
123 quotient, whole numbers, 


2464)111,35 


9856 
1279 
4 quotient, decimals in place of tenths, 
24685) 1238925 500 
123425 5... quotient in place of hund’ths. 
500 000 


A gentleman being asked how much money he had, said, 
in his pocket and purse he had $528,75, but the sum he 
had in his pocket was the square root of what he had in 
his purse. How much had he in his purse ? 

Ans, $22,50. 


PROBLEMS, 191 


PROBLEM XIII. 


To find the involved square root of ris fee: number, 
the square of which shall be equal to 4 3 &e. of the 
remainder given. 


Rule. Wivide, the denominator by the nunledatce: multi- 
ply the given number by the quotient, extract the involved 
square root of the product, which being divided by the 
number which the given number was multiplied, the quo- 
tient will be the root required. 


Nore. If it be required that the square of the involved 
root be 2, 3,'4,; &c. times larger that the remainder, the 
given sum must be divided by the number denoting the 
fold, and tne involyed square root of the quotient must be 
multiplied by the same number. 


EXAMPLE. 


A gentleman having a piece of land 10 feet. wide and 
20 feet long, which was too wet for cultivation, he dug a 
ditch the length of it as deep as it was wide, and within 
the 10 feet the earth being thrown on the remaining 
ground raised it a half foot. What was the size of the 
ditch ? 

12 10 the width of the ground, 
ior 2 
2 — 
20(4 involved root, 
16 


2)4 


2 ditch 2 feet square. 


olsnl 


PROBLEM XIV. 


If it be required to find the side ofa square which is in 
depth to its width as 2 to 3, 3 to 4, 4 to 5, &c. within any 
given number which square shall be 13 2 3 &c. or 2, 3, 4, 
&c. times as large as the remainder of the number given, 


Rule. Multiply or divide the given number, as the ques- 
tion may require, as directed by problem 13th; state as 


192 PROBLEMS, 


the width required is to the depth, so is the product or 
quotient to afourth number. ‘The involved square root 
‘of that fourth number will be the depth required ; then as 
the proportioned depth is to the proportioned width, so is 
the actual depth to the actual width. | 


EXAMPLE. 


There is a piece of land 42 feet wide and 100 feet long 
to be ditched its length and within its width. The depth 
of the ditch to its width as 3 to 4, and so large that the 
earth thrown from it will raise the remaining ground two- 
thirds of a foot—what is the size of the ditch ? 


Ans. depth 4 feet, width 6. 


2)3 Width. ~ Depth. 
-- as 3 is to2 so is 63. 
1,5 den’r of number, 2 
42105 | 3)126 
1,5 width of ground, © 1,5)6,0 —-- 
Aus | a3 42)6 
210 Ans, .4::depth, 36 
42 . ay 
ee r.D, #: W..1.D..-6 
§3,0 2 = -3-- -4 6 quotient 
: 2 er 
— ‘0 
2)12 


Ans. 6 feet wide. 


PROBLEM XV. 


To find what number by being squared will increase 
1,-2,. 3,54; Ge 


Ruxz. Extract the involved square root of the number 
denoting the increase, to which root add unity or 1, and 
the sum will be the number required. 


PROBLEMS. 193 


EXAMPLE. 
What number by being squared will increase 6 1 


Increase. 
6 {2 
4 1 unity, 
2 3 Answer. 
2 quotient square rodt. 


PROBLEM XV1. 


When two numbers are given, increasing equally alike, 
to find when the smaller number will become the square 
root of the larger. 


Ruue. Subtract the smaller number from the larger 
number; extract the involved square root of the remain- 


der; to which add unity, and that sum will be the smaller 
number. 


EXAMPLE. 


N. is 34 1-2 years old when B. is 4 1-2 years old, what 
will Bs age be when A.’s age is the square of his ? 


34,5 

4,5 
31,(5 root, 
25, unity, 


ae 


~& 
5. 6 years, Amswer. 
5 quotient, square root. 


A man being asked how much money he had, said, 
extract the square root from the sum I have ; the remain- 
der will be $21,84 cents. How much money had he? 


Ans. $27,04. 
PROBLEM XVil. 


When the sum of the squares of two numberts is given, 
one the square root of the other, to find those numbers. 
18 


' 194 : PROBLEMS. 


Ruiz. Extract the involved square root, which will give 
the larger number, and the square root of that will be the 
smaller number. 


EXAMPLE. 


The sum of the squares of two numbers is 20, one the 
square root of the other: Numbers required. 
Answer: 2 and 4. 


(2 smaller numbe# 20(4 larger number, 
16 


es 


4 
4 quotient. 


0 


4 
4 
0 


The square of the hypotenuse of aright angled triangle 
is 90 feet, the base is the square root of the perpendicu- 
lar. Required the length of each, 

Ans. : base 3, perpendicular 9. 


PROBLEM XvVi1ll. 


When thessum of the squares of two numbers is given, 
and the sum of the squares twice the product, and sum of 
the numbers also given, to find the numbers. 


Rue. From the sum of the squares twice the product, 
and sum of the numbers, extract the involved square root, 
which root will be the sum of the numbers ; then by Pro- 
blem 3d find the numbers. * 

* 
EXAMPLE. 


The sum of the squares of two numbers is 20; the sum 
of the squares twice the product, and the sum of the num- 
bers is 42.. What are the numbers? 


[Solution, next page. | 


PROBLEMS. 195 


42(6 sum of the numbers, 
36 


a 2)20 square of sum of numbers, 
6 cies 
6 quot nt. 10 half square. 
9 


0 ioe 
1 difference and square root. 
2)6 half sum of numbers, 


2 and 4 Answer. 


The square of the hypotenuse of a right angled triangle, 
after deducting the base and perpendicular, is 32, and the 
length of the base and perpendicular added to twice the 
area, the sum will be 29, What is the length of the base 
and perpendicular ? Ans. : base 4, perpendicular 5, 


Note. The 32 is the square of two numbers less the 
sum, the 29 is the product of two numbers more the sum ; 
29 and 32 added is 61, and is the sum of the square and 
product of two numbers ; and the 29 added to 61 the sum 
will be the sum of the squares, twice the product and sum 
of the numbers. 


Lemma Ist. If the square root of any number be multi- 
plied, the product will be the square of the square multi- 
plied by the square root of the number by which the root 
was multiplied. 


Explanation. Let 100 be the square and 10 its root; 
10 multiplied by 4 is 40, which is the square root of 1600, 
or of 100 multiplied by the square of 4. 


Lemna 2d. If the square root of any number be divi- 
ded, the quotient will be the square root of the square 
divided by the square of the number by which the root 
was divided, 


Explanation. Let 100 be the square and 10 its root; 10 
divided by 4 is 2.5, which is the square root of 6.26, or 
of 100 divided by 16, or the square of 4. 


196 PROBLEMS. 


Lemma 3d. If anumber with 2, 3, 4, &c. times its root 
be involved, be divided by the square of the number by 
which its root is involved, the quotient will be the number 
With its root. 


Explanation, Let 100 be the number, and 20, twice its 
root, added together, will be a number with twice its root 
involved ; 120 is a sum with twice its root involved, divi- 
ded by 4 (which is the square of the fold of the root,) 
gives a quotient of 30, which is a number with its root ; 
then by Problem 12th the root is found. 


Lemma 4th. If a number is involved with the square 
root of 2 or 3 times its number, divide said number by the 
figure denoting said fold, and the quotient will be a square 
with its root; then by Problem 12th the root is found. 


Explanation. Let 100 be a number, therefore 20 will be 
the square root of 4 times its number, consequently 120 
will be a number with the square root of 4 times its num- 
ber involved. 


Question. A man being asked how much money he had, 
said, multiply the sum by 4, the square root of that pro- 
duet added to the sum I have} will be $440. How much 


money had he ? Answer : $400. 
4)440(10 
110 a number with its square root involved, 
1 10 
— 4 
20)10 wy 
10 quotient. 40 involved root, 
10 i 
100 
4 


100 Answer. 


_ PROBLEMS. 197 


Note. 440 is anumber; one part multiplied by 4 is equal 
to the other part required. 


Lemma dth. Any two numbers that are in proportion 
as one is one, and the involved square root of one. The 
difference of their square is equal to their product. Any 
two numbers whose difference is 1, 2, 3, &c., the differ- 
ence of their squares will be 1, 2, 3, &c. times their sum, 
and any two numbers whose difference is the invol ved 
cube root of one ; the difference of their cubes is equa to 
the sum of their squares: 


Required two numbers, the difference of whose squares 
is equal to their product, and their product equal to 3 times 


their sum. Answer: 4,854101 and 7,854101. 
: Dak No. 
If ,618043 require 1 
Diff. 


what will 3 require ? 
Answer : 4,854101, to which add 3, the sum will be 
7,854101, the other number. 


Required two numbers, the difference of whose squares 
is equal to their product, and the difference of their cubes 
equal to the sum of their squares. 


Divide 100 dollars between A. and B., let A.’s part be 
to B.’s as B.’s is to 100. 


Note. As the product of the first and third of 3 num- 
bers, that are in proportion to each other, is equal to 
square of the second number, it is obvious that the 100 
dollars must be so divided that one part squared must be « 
equal to the other multiplied by 100, or is 4 number with 
the root of 100 times its number involved. See the rule 
by Lemma 4th. 


[Solution, next page.| 


198 PROBLEMS. 


100)100 
100(0,618034 involved root of 1, 
36 
abe | 1 
64 0,618034 square root, 


6 quotient, — 
os »381966 square, 
121) 400 100 
121 ee 
———e 38,196600 A.’s part, 
279 61,803400 
1 quotient, 


1228)17900 618034 
9824 100 
8076 61803400 
8 quotient, 
123603)760000 
sae, 370809 
389191 
3 quotient, 


1236064(8919100 

4944256 

3974844 
4A quotient. 
A. and B. bought a farm of 300 acres; at $2 per acre, 
each paying equally; a brook ran through the farm, divi- 
ding it into two parts equally valuable, but the western 
part was worth six shillings per acre more than the east- 
ern: how many acres were there in each part, and what 

was the land worth per acre ? 


wer 


PROBLEMS. 199 


Answer : The western part contained Acres, 
122,79979 
Shillings. 
and was worth - OR pak ” 19,544003 
Acres. 
The eastern contained - “ - 177,20021 
. Shillings. 
and. was worth wei - - 13,544003 


PROBLEM X1X 


The sum of the three numbers which are in arithmetical 
proportion, and the sum of their squares given to find the 
numbers. 


Rue. Divide the sum of the numbers by three, the 
quotient will be the second number, which, being sub- 
tracted from the sum of the numbers, the remainder will 
be the sum of two other numbers, and being squared and 
subtracted from the sum of the squares, the remainder 
will be the sum of the square of the two other numbers, 
which numbers are found by Problem 3d. 


EXAMPLE. 


The sum of three numbers which are in arithmetical 
proportion is 12, the sum of their squares is 66: what are 
the numbers ? Answer: 1, 4 and’ 7. 


3)12 sum of the numbers, 
4 second number, 
4 66 sum of the squares, 
oo 16 
16 square, — 
50 sum of the square of the 
other two numbers, which are found by Problem 3d. 


PROBLEM XX. 


If the sum of 1, 2, 3, 4, &c. time, the third and first 
numbers added to the sum of the numbers, be given with 
the sum of the square of the numbers, for every addition 
add two to three, by which sum divide the given numbers 


200 PROBLEMS. 


and the quotient will be second number. ‘Then proceed 
as above, and twice the second will. be the sum of the 
other two numbers. 


Note. The second number of any three numbers which 
are in arithmetical proportion is one-third of the sum of 
the numbers. 


There are three numbers in arithmetical proportion, the 
sum of the numbers, with the addition of the first and third 
numbers, is 15, and the sum of the squares is 29. What 
are the numbers ? Ans. 2, 3 and 4. 


QUESTIONS, 
SOLVED BY RULE OF THREE DIRECT, OR INVERSE. 


There is a cistern which has a stream of water running 
into it; it has ten cocks ; all running together will empty 
it in 2 1-2 hours, 6 will empty it in 5 1-2 hours. How 
Jong will it take three to empty it? ©=Answer: 55 hours. 


‘Note.—The 6 cocks will discharge in 4 1-6 hours what 
the 10 cocks will in 2 1-2 hours, therefore it would take 
the 6 cocks 1 1-3 to discharge what would run into the 
cistern in 3 hours, therefore it would take the 6 cocks 1,111 
to discharge what would run in in 2 1-2 hours; conse- 
quently, 2 2-3 cocks to discharge the water as fast as it 
run in. 

There is a stick of timber 12 feet long, to be carried by 
3 men; one carries at the end, the other two carry by a 
lever; how far must the lever be placed from the other 
end, that each may carry equally ? 3 

Answer: 3 feet from the end. 


Note. All bodies gravitate in an inverse proportion to 
the distance of the centre of gravity. 


PROBLEMS, 201 


As 1 is to 6, the centre, so is 2 to the answer required. 


[0 RS RR ee baleen» 
1 
2)6 

3 Answer. 


A stick of timber 30 feet long, to be carried by 5 men, 
two carry at one end, the otherthree by a lever; how far 
‘from the centre must the lever be placed that all may carry 
equally 4 


M. F, M. 
Ar Ge Os se TERS) ees 
2 
3)30 
10 Answer. 


A. and B. carries a stick of timber 30 feet long ; A. car: 
ried 8 feet from the centre of the stick, and B. 10 feet; 
what part of the stick does each carry ? 


A. 8 Aen. aed 28 
_B.10 *27 BTS 1-3 
TS a 80 aa ag TR 80 nO 
3 10 
18)240 18)300 
13 1-3 feet. 16 2-3 feet. 


If 4 acres will pasture 40 sheep 4 weeks, and 8 acres 
will pasture 56 sheep 10 weeks, how many sheep will 20 
acres pasture 50 weeks ? Answer: 108 sheep. 


Note. 32 sheep will eat the pasture of 8 acres in the 10 
weeks, provided the grass had not grown but 4 weeks, 
therefore it took 24 sheep 10 weeks to eat what grew in 6 
weeks, therefore it would take 40 sheep to eat the grass 

is* 


202 QUESTIONS FOR EXERCISE. 


that will grow on 8 acres; consequently, the grass grows 
on each acre sufficient to keep 5 sheep, and the pasture on 
each acre is sufficient to keep 20 sheep one week. 


Note. The grass is continually growing. 

A. and B. carries a stick of timber 30 feet long; A car- 
ries 8 feet from the centre, and carries 16 2-3 feet: how 
far from the centre must B. carry to carry the remainder? 

Ans. : 10. 
feet. ~ feet. feet. 
As 16,666 isto 8 so is 13,333 to 10 Answer. 


QUESTIONS FOR EXERCISE. 


- 1. If 42 oxen will eat 34 acres of grass in 4 weeks, and 
21 oxen will eat 10 acres in 9 weeks, how many oxen 
will eat 24 acres in-18 weeks, the grass standing equal 
on every acre and growing uniformly? Ans. 36 oxen. 

2. What distance from the corners of a square stick of 
timber, a side being 28 inches, must lines be drawn for 
the purpose of hewing it 8 square or an octagon ? xy 

Ans. 8,201 inches. 

3. P’s farm is a mile long and of an equal breadth ; Q’s 
farm is a square, containing the same quantity of land as 
P’s; deduct the breadth of P’s from its length, and the re- 
mainder will be the length of a side of Q’s, which side is 
required ? Ans, 197,77 rods. 

4, There are 4 wheels taking hold of each other, the 
first has 33 teeth, the second 34, the third 36, and fourth 
38, how many revolutions will each wheel have made 
when they shall all come to the same teeth where they 
began ? 


Ist =3876) 
4 pa . Revolutions. 
4 3366 5 


5. A lets B have 1000 lbs. of live sheep on the condition 
that B doubles the weight at the end of 4 years, but A 
requests his just proportion at the end of two years. 
Query, what weight shall A receive ? 

Ans. 1414, 21356-+ Ibs. 


QUESTIONS FOR EXERCISE. 203 


6. A sugar loaf of a conical form, whose slant heighth 
was 15 inches weighed 14 lbs. [want to cut off 3 Ibs. 
from the top of this loaf by a plane parallel to the base, 
at what distance from the top, measured on the slant 
heighth, must this section be made ? Ans. 8,97 inches. 


7. Suppose three men, A, B and C, to travel round a 
course of 5 miles in circuit, A at the rate of 6 miles per 
hour, B 5, and C 4 miles per hour, at what distance must 
they be placed from each other at starting, so that travel- 
ing the same way, they may all come together in 21 hours? 

Ans. 1 mile apatt. 

8. A grazier bought in as many sheep as cost him £60, 
out of which he reserved 15 and sold the remainder £54, 
and gained 2 shillings a head by them, how many sheep did 
he buy, and what did they cost him per head ? 

Ans. 75 sheep, 16 shillings per head. 


9. A wagoner drove a certain distance with his empty 
wagon, at the rate of three miles per hour, and returned 
with a load at the rate of 2 miles per hour, accomplishing 
the whole journey in 25 hours. What distance did he 
travel ? Ans. 30 miles. 


10. Viewing through a telescope a brick house at a dis 
tance, I observed that the telescope took in exactly 17 
courses of the brick work, I then measured 90 yards 
directly towards the house and found that the field of the 
telescope now took in only 12 courses. Required the dis- 
tance of the house from each place of observation. 

Ans. 306 yards from the first place. 


11. The area contained at the centre between six equal 
circles is ten acres. Required the diameter of the smallest 
circle which will just enclose the whole. 

Ans, 29,6 chains. 

12. There are two pillars in a straight line, perpendicu- 
lar to the plane of the horizon, whose distance asunder is 
180 feet, the one is 60 and the other 40 feet high; in what 
part of the line of distance a ladder may be fixed so as to 
reach the top of each pillar, without moying its bottom ; 


also the length of the ladder? 
Answer, 95,55 feet from lesser. 


103,589 feet ladder. 


204 QUESTIONS FOR EXERCISE. 


13. A man that in stature is just six feet high, 
From his lowermost parts to his eye, 
Stands straight on a globe in diameter meet, 
That’s found just to measure 100 feet ; 
Suppose that the ball in the air is suspended, 
And just half a mile from the earth is extended, 
The number of acres of earth there below, 
That is hid from his view I desire to know. 
N. B. Hang mile to the centre of the ball. 
Ans. 2060, 4327--acres. 


14. Twomen travel round a course of 4 miles in circuit; 
when they both set off together and travel the same way, 
the one gains a round of the other in 10 hours; but when 
they travel contrary ways, they meet every hour; at what 
rates respectively do they travel ? 

m. m. 
Ans. 2,2 and 1,8 per hour. 


15. The sum of two numbers is 1011, the cube root of 
3 of the quotient of the greater divided by the less is 7 ; 


what are those numbers ? Ans. 1009,;4°%. greater. 
1, $242 less. 


16. Divide $1000 among 3 men, giving the first one- 
third more than the second wanting $10, and the third half 
as much as the second and $20 over; I demand each 


man’s share. Ans. A’s share, $455435 
B’s “e 34955 

C’s «19432 

Proof, 1000 


me A, B, and C company and put in together £3860, 

A.’3 money was in 3 months, B.’s 5 months, and C.’s 7 

months ; they gained £234, which was so divided, that $ 

of A.’s gain was equal to 4 of B. ’s, and + of B.’s equal to 

1 of C.’s ; what did each gain and put in? 
& 


L 
: A. gained 52 and pat in ©1400 
Ans Beef 18a" Pea 260 
CoS 104° eae 1200 


18. A person has a circular yard, 150 feet in diameter, 
and wishes a gravel walk of equal width, made round it 


QUESTIONS FOR EXERCISE. 205 


within the fence. Required the width of the walk, so 
that it may occupy a fifth part of the ground. 
| Ans. 7,918 feet. 
19. A gentleman has a garden 100 feet long, and 80 
feet broad, and a gravel walk is to be made, of equal width, 
half round it; what must be the width of the walk, so 
that it may take up 1 of the ground ? 
| Ans. 11,8975 feet. 
20. A, B, and C sold 300 yards of cloth for $900, and 
each sold to the amount of $300; B sold his for one 
dollar per yard more than A, and C for one dollar more 
than B; how many yards did each sell ? 


A. sold 135, 481-++ 
Ans} Bet 93, 332+ > Yards. 
Cc. « 71, 186+ 


21. Required the dimensions of an oblong garden, con- 
taining 3 acres, and bounded by 104 rods of fence. 
Ans. 40 rods by 12. 
22. Given the slant height of a right cone, standing 
perpendicularly on a horizontal plane, in latitude 540 36’ 
North—20 and the diameter of its base—12 feet, to find 
the time at which, onthe 24th of June, the area of the 
visible part of its shadow is equal to the part of its curved 
surface which is not enlightened by the direct rays of the 
sun. Ans. 56 minutes past 6, A. M. 
or 4 , th dices Wan 


23. What is the diameter of a solid globe of glass, 
which when blown into a hollow globe, until the shell is 
but + of an inch in thickness will be sufficient to contain 
ten gallons of wine ? Ans. 7, 459 inches. 

24. The area of a given trapezimm is 20 acres, the 
diagonal is 5 chains longer than the sum of the perpen- 
diculars, which are in proportion of 3to 5. Required the 
dimensions. Ans. 22, 6556+. 

25. Charles, Henry and William having a quantity of 
chesnuts, made the following propositions, viz., says 
Charles to Henry and William, give me one third of your 
chesnuts and I shall have 100. Henry says to Charles and 
Wiiliam, give me half of yours and I shall have 100. Says 
William to Charles and Henry, give me one-quarter of 


‘ 


Now, I would ask, how hymen could contrive 


206 QUESTIONS FOR EXERCISE. 


yours and I shall have 100. How many Chesnuts had 
each boy ? 
Charles had 643¢ 
Ans. < ‘Henry “ 292+ 
William “ 762+ 


26. The sides of a triangle are 32, 40, and 60 rods; 
how far from each corner must a house be placed to be 
equally distant from the corners ? Ans. 64, 9 rods. 

27. A merchant purchased 63 galls. of wine, for $100, 
but_by a.Jeak in thé cask a certain quantity was lost ; he 


_ then sold the remainder for the original cost, and gained 


a sum per cent. equal to twice the number of gallons lost. 
Required the number of gallons lost. Ans. 13 gallons. 

28. Three merchants join stocks together. The first 
man’s stock was less than the second man’s by £13 ; the 
second and third man’s stock was £175. In trading they 
gained £48 more than their whole stock; the first man’s 
proportional part of their gain was £78; what was each 
man’s stock and gain ? 


Av’sstock £65 gain £78 
Ans. < B's 78 es 93, 12s. 
C's « 97 * 116, 8s. 


29. THE PARADOXICAL REEL. 


The grandsire with the granddame first the reel began, 
l'wo fathers and two mothers followed on, 

T'wo brothers and two sisters joined the dance, 

Two husbands with their wives did then advance, 
T'wo.uncles and an aunt the next appear, 

With two sons and a daughter in their rear, 

Iwo cousins with a nephew and a niece, 

And a young grandson clostd the fancy piece. 

Soon as the dance was through, they counted o’er, 
And five were all the persons on the floor. 


To make this number count full twenty-five ? 


™~ 


EXPLANATION. 


A gentleman married a widow who had a daughter, 
afterwards his brother, (a widower,) who had a son 


QUESTIONS FOR EXERCISE, 207 


married the daughter; from these five the foregoing rela- 
tions are made out. 

30. Four men have each such a sum of money, which 
being put together makes 250s, and if to the first man’s 
money be added £8, it will be just as much as the second 
man’s money décreased by £8, and as much as 8 times 
the third man’ S money, and but as much as one-eighth of 
the fourth man’s money. How much had each man? 


( A. had #1628 

ad Bot Baas 
ns. } C. « Byles 
De Sore 


Proof. £250 


31. A gentleman owning a farm in the form of a circle, 
whose diameter was 60 chains, in his will gave his six 
daughters the six largest equal pentagons that could be 
formed with an angle of each touching the periphery of 
the circle ; to his son the largest hexagon that could be 
formed about the bars of the pentagon, and the remainder 
tothe widow. How many acres had each 4 


Sons, 40, 429 
A Daughters 26,876 
Widow, 108, 485 


32. Required a whole number to whichif 8, 19, 32, 47 
and 64 be severally added, each sum shall be a square 
number universally. Ans. 17 

33. Several merchants enter into partnership, each one 
put into the stock 65 times as many pounds as there were 
partners; with that stock they traded and gained as many 
pounds per £100 as there were partners. Now, if £10, 
10 shillings be added to and subtracted from ther gain, 
that sum and difference will be €6491, 6 shillings 3 pence. 
How many merchants were there ? Ans. 5 merchants. 

34. Divide $3800 in the following manner: 

My Polly and Nancy, your shares must be reckoned 

In such a proportion to Sue, 
That 5), of the first, and + ofthe second 
Will equal the third divided by 2. 


208 QUESTIONS FOR EXERCISE. 


And further, my daughters, to prove your work true, 
For Buch your proportion must be, 

That ,0f Nancy’s and + of Sue’s 
Will, equal Polly’s divided by three. 


Polly’s share, $1165, 427-++ 
Nancy’s ‘“ 1801, 117+ 
Sue’s “ 833, 45-- 


35. In a triangle containing 100 acres, and the sides 
equal, the distance from a stoke in the triangle is in this 
proportion, viz., If 1 of the two greatest distances were 
added to the shortest, it would make a certain number of 
chains ; if} of the greatest and least were added to the 
other, the sum would be the same ; if 4 of the two shortest 
were added to the greatest, the sum would be the same. 
The distance from each angle to the stake is required. 


30, 55 
u Ans. 5 28, 34 > Nearly. 
24, 66 


36. Having evacuated three pieces of canal, containing 
in the whole 3000 yards; I have 64 cents per yard for 
the first, 7 cents for the second, and 74 cents for the third. 
When I received payment, I found the pieces amounted 
to equal sums of money ; how many yards did each piece 
contain ? 


2d “ 996, 592 
3d“ “ 930, 153 


37. There are four wheels of the following dimensions, 
viz: The first 10 feet in circumference, the 2d 20, the 
third 30, and the fourth 40; they each have a black spoke 
on which they stand. It is required to know how far each 
wheel shall roll, to have them all stand on the same black 
spoke, as at the beginning ? Ans. 120 feet. 

38. Ifa solid globe of glass at the furnace, whose diame- 
ter is 8 inches, be blown into a hollow globe, till the shell 
is but + of an inch in thickness, what will then be its 
diameter, and how much wine will it hold ? 

Ans. 20,655 inches diameter. 
18,83 gallons. 


lst piece contained 1073, 253 
Ans, 


QUESTIONS FOR EXERCISE. 209 


39. On the 4th of July a pole was erected, 
Composed of six pieces, and nicely connected ; 
Two feet and six inches it measured around, 
At the place where it stood on the top of the ground. 
The form was a cone, in surface complete; 

The heighth of the same was twice 60 feet. 
What length of inch ribbon procured at the shop, 
Will wind round the pole from bottom to top, 
And have it lay smooth, and plain to be seen, 
By leaving a space of 5 inches between ? 
Ans. 300 feet. 
40. A gentleman owning a farm in the form of a circle, 
bequeathed to his five sons, the five largest circles that 
could be formed within, and touching the periphery of the 
circular farm. To his daughters he gave that portion of 
land at the centre contained between the peripheries of 
the five equal circles, which was 40 acres, and the remain- 
der to the widow. I demand each son and widow’s share. 
Ans., son’s, 57,92 acres: 
widow, 93,05 “ 
41. A gentleman at his decease left a widow, son and 
daughter, and annexed to his will was a sealed packet 
with directions for it not to be opened till the son and 
daughter had arrived at lawful age. When they were of 
age, the packet was opened, and found to contain $1000, 
to be disposed of as follows; If the son should marry 
before arriving at age and the daughter not, he was to 
have three parts and the mother one. If the daughter was 
to marry before arriving at age and the son not, the mother 
was to have three parts and the daughter one. It so 
happened that both son and daughter married. Query, 
how is the $1000 to be divided ? 
Son 9 parts equalling, $692;4 
an.) Mother 3 “ ‘s 23012 
Daught.l “ oo 6712 


3 


$1000 Proof. 

42. How many head of cattle may come from a heifer 
calf in 20 years, provided they are all of the feminine 
gender, and each one to have a calf when she becomes 3 


years old, and after that have a calf every year? 
Ans. 1278 head. 


210 QUESTIONS FOR EXERCISE. 


43. The product of two numbers is3802,5; they are in 
the ratio of 8 to 5; what are those numbers ? 
Ans. 78 and 48,75. 
43. Required the diameter of a circle which shall just 
enclose six of the largest equal circles that can be formed 
within it, and have 50 acres about the centre between the 
peripheries of the equal circles. Ans, 41, 6168 chains. 
44. In an oblique angled triangle there is given the pro- 
duct of the two sides 186, their difference 3,5, and the 
shortest side is to the base inthe ratio of 4 to 7. The 


sides are required. 15,5) - 
; Ans. 2 12 Sides. 
Lee 
7 


45, There are two numbers, the product of whose 
multiplication is 79,625, and that of their cubes is 
2112, 890625. ‘What are the numbers ? 

Answer, 6, 5 
| 12, 25 

46. The diameter of a circle is 40 chains. Required the 
length of the sides of ten of the largest equal pentagons 
that can be formed within it, with an angle of each touch- 
ing the periphery of the circle, and leaving a decagon at 
the centre, whose side shall be equal to the side of each 
pentagon ? Ans. 6, 5 chains. 

47. A gentleman has a garden in the form of an equi- 
lateral triangle, containing half an acre; the soil being 
low and naturally moist, wishes to raise it, by heaping 
earth upon it. How wide and deep must a ditch be dug 
on the outside of the triangle, but within the limits of the 
half acre, to yield earth sufficient to raise the ground plot 
one foot, the width and depth being equal ? 

Ans, 5 feet 4 inches. 

48. Says A. to B.,if I had 5 of your crowns, I should 
have twice as many as you would have left; and says B. 
to A., if I had 3 of yours, I should have four times as many 
as you. How many had each ? 

Ans. A. had 62, B. had 103. 

49. A country spark addressed a charming she, 

In whom all lovely features did agree ; 
But he not skilled, as you may now presage, 
Was too solicitous to know her age. 


QUESTIONS FOR EXERCISE, 211 


The lady smiled at this preposterous rule, 
But condescends to satisfy the fool ; 
Made him this answer, with a generous air, 
A lofty smile, peculiar to the fair : 
My age is such if multiplied by three, 
And 2 of that product trebled be, 
The square root of 2 of that is four ; 
So fare ye well, you are to know no more, 
Your fond impertinence has caused this rage, 
Tis clownish, sir, to ask a woman’s age. 
Ans. 28 years. 
50. A gentleman bequeathed to his six sons and widow, 
a circular piece of land, containing 785, 4 acres, in the 
following manner, viz. : to his eldest three each the largest 
circle that can be inscribed within the circular tract ; to his 
widow, the piece at the centre and bounded by the three 
equal circles ; the three remaining pieces to his younger 
sons. I demand each ones’ share. 


Eldest sons’ share, 169, 167716 
A Widow’s share, 8, '702423 
Younger sons’ share, 89, 731476 


ols general forming his army into a square, finds he 
has 284 soldiers.over and above a square; but increasing 
each side with one soldier, he wants 25 to fill up a square. 
How many soldiers had he? Ans. 24000 men. 
52. Two men, A. and B., purchased 200 oranges, and 
gave $10 for them, in the payment of which each paid $5. 
Now, says A. to B., give me my choice in the oranges, 
and I will allow 2 cents per orange more than you do. 
Required how many each had, and at what price per 
orange ? Hh } A. had 80, '7417-+at' 06,192-++ 
“UB. “119, 2582-++at 04,1924 


53. A. B. and C. purchased a block of land, containing 
4000 acres, at $3,75 per acre; a canal runs through the 
S. E. section of it; and A., willing to avail himself of the 
accommodation thereof, proposes to B. and C. to pay 75 
cents per acre more for this section than they, provided 
they will allow him his proportion in it; agreed. Now, 
before B. and C. make a division ; a rail road has been . 
commenced through the N. W. part of the block, in con- 


231 4 QUESTIONS FOR EXERCISE. 


sequence of which, B. proposes to C. to pay $1,25 per 
acre more than he for his choice, (agreed also.) Now, 
admitting that each paid one-third of the original purchase, 
and none give or receive cash afterwards, how must the 
land be divided to answer these conditions, and what will 
each man’s proportion stand him in per acre ¢ 
Ho) iP 
A. 1167—3—31, 8 at $4, 28,1 
Ans. < B. 1172—3—5,76 “ 4, 26,335 
C. 1659—1--2,44 3, 01, 335 
4000 

54. What is the side of that equilateral triangle, whose 
area costs as much for paving, at 8 pence sterling a foot, 
as the pallisading of the three sides did at a guinea per 
yard ? | | Ans. 72, 746 feet. 

55. A captain gives to one of his men a plank 9 by 16 
_ inches, with orders to make a lid 12 inches square, and 
only to cut the plank in two pieces. The man being igno- 
rant of lines, would be obliged to any one to draw a plot, 
how the plank is to be cut. 


Lee pe pod 


56. It is required to find the thickness of the lead in a 
pipe of an inch and 4 bore, which weighs 14 lbs. per yard 
in length, the cubic foot of lead weighing 11325 ounces. 

Ans. 575345 inches. 

57. Having surveyed an isoseles triangle, whose equal 
sides are ten rods each, I find if I either add or subtract 
2 rods to or from the other side in each case, the area will 
_ be 2 rods Jess in the calculation than the true content. 

Required the side. Ans. 14 rods. 

58. If a heavy sphere, whose-diameter is,4 inches, be 
let fall into a conical glass full of water, whose diameter 
is 5 and altitude 6 inches. It is required to determine how 
much water will run over. Ans. 26,272 cubic in. 

59. The dimensions of the sphere and cone being the 
same as in the last question, and the cone only + full of 
water. Required what part of the axis of the sphere is 
immersed in the water. _ Ans. ,546 pts. 


QUESTIONS FOR EXERCISE, 213 


60. The cone being still the same, and 1 full of water. 
Required the diameter of a sphere which shall be jus 
covered by the water. Ans. 2, 445+. - 


61. A certain man disposed by will 
A. circular piece of land, 
To his dear wife and daughters three, 
As here below doth stand. 
The circle was precisely such 
When measured through the centre, 
By Gunter’s chain it did contain, 
In numbers four and twenty ; 
His daughters’ portions were alike, 
In the*aforesaid ground ; 
They each must have a circular piece, 
As large as can be found. 
And the remainder of the land 
Unto his wife he willed, 
And left it with a faithful hand, 
To see it all fulfilled. 
At £20 an acre just, 
This land must valued be. 
Now tell me what’s the widow's worth, 
Likewise the daughters three. 


Acres. 
‘Ana Daughters’ share, 29, 232, equals £584, 640 
" \. Widow’s 6 6 EG OO LUE 48 a cay LA0s 4 


62. Supposing there is a circle 20 feet in diameter, how 
large a trench of equal breadth and depth will it take to 
fill this circle in the form of a half globe, the trench to be 
dug around the circle? Ans. 5, 148689-+feet. 

63. If6 oxen or 10 colts can eat 21 acres of pasture in 
14 weeks, and 10 oxen and 6 colts can eat 45 acres of 
similar pasture in 20 weeks, the grass growing uniformly, 
how many sheep will eat 240 acres in 40 weeks, admitting 
that 1134 sheep can eat the same quantity as 12 oxen and 
22 colts % Ans. 3472 sheep. 

64. If 9 gentlemen or 15 ladies will eat 17 apples in 5 
hours, and 15 gentlemen and 9 ladies can eat up 47 apples 
of a similar size in 12 hours, the apples growing uniformly, 
how many boys willeat up 360 apples in 60 hours, admit- 


214 QUESTIONS FOR EXERCISE. 


ting that 120 boys can eat the same number as 18 gentle- - 
men and 26 ladies ? Ans. 642--boys, 

65. A certain house standing on a plain is 28 feet wide, 
and 35 feet high, the top of the roof being 15 feet higher 
than the feet of the rafters. It is required to find the 
length of a ladder that will just reach from the ground to 
the top of the house, and lie on the roof exactly parallel 
to it. | Ans. 47, 8-+-feet? 

66. A note was given for $35, dated 3d day of 4th 
month, 1827, payable one year after date, and the condi- 
tions of payment were these: provided one-half of the 
value of the said note be paid 6 months after date, that 
the remainder should become due 18 months after date, 
and so in proportion to the value of the payments and the 
time when paid. On said note were the following pay- 
ments, viz., $10 10 days after date, $10 the 2Ist day of 
7th month, 1827, and $5 the 8th of third month, 1828. 
Required the time after the date of the note, that the 
remainder shall become due. Ans. 9883 days, 


67, A landed man two daughters had, 
And both were very fair ; 
He gave toeach a piece of land, 
One round, the other square. 
For 20 shillings an acre just, 
Each piece its value had ; 
But when they parted with the same - 
It made their hearts full glad ; 
For, by a contract fairly done, 
The price of each was made, 
The shillings that encompassed each, 
Foreach exactly paid. 
If across a shilling be an inch, 
The which is very near, 
The query is, which sold for most, 
The round piece or the square ? 


Ang, § Round piece, 3941225,+shill. 
; { Square piece, 5018112,---shill, 


68. How deep must I saw into a round log, 36 inches 
in diameter, to cut it 3 off? Ans. 13,-+inches. 
69. Suppose a crown that shall weigh 60 Ibs. is to be 


QUESTIONS FOR EXERCISE. 215 


made of gold, brass, iron and tin, mixed together in such 
proportions that the weight of the gold and of the brass 
together may be 40 lbs.; the joint weight of the gold and 
of the tin 45 lbs., and the joint weight of the gold and of 
the iron 36 lbs. How much of those four metals must be 
taken ? 

( 304 pounds gold, 

iG gaat Save Dress; 
5d {> Iron; 


LAS (882: cts 


_ 70. Suppose two towers standing on a common level ; 
the one is 280 feet and the other 210 feet high; the whole 
distance between the towers 225 feet. I would know the 
length of a ladder that will reach the top of each without 
moving the foot, and also the distance of the ladder from 
the foot of each tower. 

. 282, 34 Jength of the ladder. 
Ans. { 188, 72 from one tower. 
36,28 from the other. 


71. The discounting of a bill which at a certain rate per 
cent. per annum came to £5, 12 shillings, and would at 
one per cent. more have cost £6, 6 shillings, and at one 
per cent. less no more than £4, 18 shillings. The value 
of the bill, time, when due, and rate of interest is required. 


and time are any two numbers whose 


Rates of interest 7, 8, 9, and the price 
Ans. 
product equals 1400 shillings. 


72. A man hada stone which he kept as a weight for 
his scales, which weighed 1093 lbs. ; he broke it into 7 
pieces, after which he could by its assistance not only 
weigh 1093 Ibs , but every pound under 1093. What was 
the weight of each piece ? 

Ans. 1, 3, 9, 27, 81, 243, 729,=1093. 


73. Required two numbers, such that the lesser number- 
may be contained in the greater twice, with a remainder ; 
this remainder ir the lesser twice, with a remainder; this 
new remainder in the first remainder twice, with a re- 
mainder, and so on, every remainder to be found twice 


216 QUESTIONS FOR EXERCISE. 


with a remainder, till the eleventh division, in which the 
remainder is to be a rational cube. 

ei pide 70693 
*) Dividend 170668 


74. To pay a certain number of workmen at the rate of 
£3 per man, €8 is wanting to the person who employs 
them; but on giving them £2 each he has £3 left. How 
many workmen were there ? Ans. 11 men. 

75. Required in what time a cistern of 200 cubic feet 
will be filled by three pipes, the first of which can fill 9 
cubic feet in 21 days ; the second 15 cubic feet in 34 days, 
and the third 19 cubic feet in 51 days. | 
Ans. 1783, days. 

76. Ifa person with an, air balloon ascends vertically 
from the city of Hudson, to that heighth that he may just 
see the top of a flag-staff hoisted on the top of the city 
hall, in New York, appear in the horizon. I demand his 
heighth above the surface of the earth, supposing the cir- 
cumference to be 25000 miles, and the distance between 
Hudson and New York 150 miles, and the flag erected 
100 feet from the ground. Ans. 2,3576 miles. 

77. A man in the centre of a circular field, containing 
ten acres, starts for the circumference, and after walking 
a certain distance, turns, making an angle of 45 degrees. 
The whole distance walked was 40 rods; how far did he 
walk before he made the turn ¢ Ans. 11, 03 rods. 

78. Three magazines containing three sorts of grain 
each; the first contains of rye 30 bushels, of barley 20 
bushels, and of wheat 10 bushels,-and costs 11, 10 shil- 
lings. ‘The second contains of rye 15 bushels, of barley 
6 bushels, and of wheat 12 bushels, and costs £6, 18 shil- 
lings. And the third contains of rye 10 bushels, of barley 
©) bushels, and of wheat 4 bushels, and costs €3, 15 shil- 
lings. I demand the price of the rye, barley, and wheat, 


per bushel, 
Rye 4 shillings. 
Ans.< Barley 3 “ 
Wheat 5 ‘ 
79. Two travellers set out at the same time; the one 
from Dublin, and the other from Londonderry ; the former 
for Londonderry, the latter for Dublin. When they met 


QUESTIONS FOR EXERCISE. _ 217 


and had computed their journey, it was found that the 
former had travelled-30. miles more than the latter, and 
that at their rate of travelling, the former expected to 
reach Londonderry in 4 days, and the latter to reach Dub- 
lin in 9 days. The distance between Dublin and London- 
derry is required. Ans. 150 miles. 
80. Two masons, A. and B., jointly perform a piece of 
work in 12 days; now, if the sum of the days in which 
they could each have separately performed the same, be 
multiplied by the days in which A. alone (he working 
quicker than B) could have done it, the product will be 

1000. In what time could each do it? 
Ans. A. in 20, B. in 30 days. 


81. If the side of an equilateral triangle pyramid is 24 
inches, what is the diameter of the smallest globe from 
which it could be made? © Ans. 26, 91 inches. 

82. A sum of £120, 5shillings was lent on condition 
that the principal and interest at 6 per cent. per annum 
should be discharged in 7 equal payments, one at the end 
of every year, and that at each payment, the interest then 
due should be cleared, and the remainder by which such 
payment exceeds the interest, applied to. reduce the prin- 
cipal. The value of the payment is required. 

Ans. £21, 10 shill. 91 pence. 

83. Whatis the length of the side of a cubic box, which 
contains its largest. inscribed globe and one bushel to fill 
the vacancies 2 Ans. 16, 526 inches, 
. 84. A messenger being departed 9 hours froma certain 
place, travelling at the rate of 5. miles in 2 hours, another 
messenger is sent after him, who travels at the rate of 11 
miles in 3hours. It is required to find what number of 
miles the second messenger will ride before he overtakes 
the first 2 Ans. 70 2 miles. 

85. The radius of a circle is 64 chains. Required the 
length of the side of the 8 largest pentagons that can be 
formed within its periphery, leaving an octagon at the 
centre, whose side is equal to that of either of the penta- 
gons. Ans, 23, 307 chains. 

86. The age of a man, his wife and son, all added 
together, make 76 years; if the man’s age be added to his 
son’s, the sum would equal the wife’s and 20 years over, 

19 


218 QUESTIONS FOR EXERCISE. 


but if the wife’s and son’s were added, it would equal the 
father’s age and 4 years. What was the age of each 4 


‘( Son’s 12 years. 
A wes 2o. 
Father’s 36 & 


87. A. B. and C. enter into partnership. A puts in 
$100 for 14 months, and draws $112 gain. B. puts in—— 
for 12 months, and draws $128 gain. C. puts in $100 for 
——,and draws $116 gain. Required B.’s stock, and 
C.’s time. 

se B.’s stock $1172 


C.’s time 182 months. 


88. A vinter has a cask of wine containing 500 gallons, 
of which he drew 50 gallons, and then filled it up with 
water, and did so 5times. -I demand how much wine and 
water there was le{tin the cask. 


‘Avie 3 2952190 gallons of wine. 
2043527 gallons of water. 
89. The diameter of a circle is 40 chams. Required the 
length of the sides of ten largest equal pentagons that can 
be formed within it, with an angle of each touching the 
periphery of the circle, Jeaving a decagon at the centre, 
whose sides shall be equal to the side of each pentagon. 
Ans. 64 chains. 
90. Suppose the frustrum of a right pyramid to be 4 
feet square at the base, and one foot at the top, and the 
slant heighth 20 feet, and a rope 2 inches thick to be 
wound round it, so as to cover its sides from bottom to 
top. | How long is the rope # Ans. 1280 feet. 


91. A. B. and C. make a company, and put in £3860. 
A,’s money was in 3 months, B.’s 5 months, and C.’s 7 
months ; and they gained £234, which was so divided 
that 4 of A.’s gain was equal to } of 'B.’s, and. + of B.’s 
to + of C.’s. What did each gain, and put in? 

A.’s stock £1400 gain £52 
A } Bs sf 1260: «6 78 
GC. sys 1200 “ 104 


QUESTIONS FOR, EXERCISE.» 9 219 


.92...A gentleman owning atract of land.in the form ofa 
circle, containing 785, 4.acres, bequeathed,. to. his, 10 
daughters the ten largest pentagons that could be formed 
within it, with an angle of each touching the periphery of 
the circle; to. his son he,gave the largest decagon that 
could be formed about.the. basis of the pentagon, and the 
remainder to his widow. Required. each oues’ share. 

Sons’ 203, 075 
“Ans. Daughter’s 45, 409 acres. 
Widow’s 128, 235)" 

93. A vinter has a cask of wine containing 500 gallons, 
of which he drew 50 gallons, and filled up the cask with 
water; he drew the next time 40 gallons, the next time 
30, the next time 20, and the next time 10 gallons, still. 
filling up the cask with water as he drew. I demand 
how much wine and how much water is in the cask. 


3 Wine 366-2804. 
| Ans. { 31 at \ 


94. A. B. and C., with their wives P. Q. and R., went 
to the market to buy sheep; each man and woman bought 
as many sheep as they gave shillings for each sheep; A. 
bought 23 sheep more than Q., and B. bought 11 more 
than P., also each man laid vut 63 shillings more than his 
wife. Which two persons were husband and wife ? 


38=>A and R.-=31 
Ans. 12—B. “s O= 9 
B= C. = 1 


95. A tree 120 feet in heighth, stood upon an eminence, 
but being partly broken off ata certain distance from the 
ground, and the top falling down a declivity considerably 
lower than the foot of the tree, but resting upon the stump, 
the distance from the foot to the top of the tree when 
down was 90 feet, and a line drawn from the foot, ata 
right angle with the perpendicular stump, to intersect the 
part broken. off was 40 feet. Question, how high was the 
tree broken above the ground 4 Ans. 21,6981--feet. 

96. Four meu travelling together, found a purse of shil- 
lings only, out of which every one took a number at an _ 


220 QUESTIONS FOR EXERCISE. 


adventure; afterwards, by comparing their number 
together, they found, if the first took 25 shillings from the 
second, it would make his number equal with what the 
second had then left; ifthe second took 30 shillings from 
the third, his’ money would then be tripple to what. the 
third had left; and if the third took 40 shillings from the 
fourth, his money would then be double what the fourth 
had left; lastly, the fourth taking 50 shillings from the 
first, he would then have three times as much as the first 
had left and 5 shillings more. What was each person’s 
share ? 

(1st, 100 shillings. 


1 2d, 150 f 
Ans. ¢ 3d, 90 Hi 
4th, 105 5g 


97. Suppose two hollow globes to be one inch in thick- 
ness, and of such dimensions, that if the smaller globe be 
put into the larger one, it would exactly fill it; these 
globes when separate will contain 2000 cubic inches of 
fluid. Now, suppose that the exterior surface of the 
larger globe be divided by the usual number of parallel 
lines found on artipecal globes used by geographers, and 
likewise to be divided in the same manner as the globes 
are divided. The question is, what is the distance around 
this last mentioned globe at the 74 parallel of latitude ? 


inches. 
Ans, § 19:3264 Dia. of greater. 
"(13,2745 Cir. at 74 parallel. 


Rule, As radius is to Cir., so is Co Sine to the required 
circumference. 

98. There are two numbers, if the greater is added to 
its square, and from this sum we subtract the square of | 
the lesser, the remainder is 94.. But the square of the 
lesser being added to the lesser, the sum is equal to twice 
the greater. Required the numbers. Ans. 10 and 4, 

99. What must be the diameter of a vessel whose con- 
cavity is in the form of a sphere_to contain 500 wine gal- 
lons. Ans. 60,422 inches. 

100. Two porters agreed to drink off a quart of beer 
between them, at a draught each; the first drank till the 


~ 


> 


QUESTIONS FOR EXERCISE. 221 
surface of the liquor touched the opposite edge of the bot- 
tom, and then gave the remaining part to the other. What 
was the difference of their shares, supposing the cup was 
the frustrum of a cone, the depth being 6. inches, the 
diameter at the top 4 inches, and at the bottom 5 inches 4 

Ans. 15,89 cubic inches, 

101. A cooper wants a cistern, to contain 1500 wine 

gallons, the heighth to be 6 feet, and the difference of the 

diameters to be 6 inches. Required the diameters. 

Ans. 81,259 and 75,259 inches. 

102. Suppose three lots of ground, of equal area, the 

first a pentagon, the second a hexagon, and the third a 

heptagon; each lot is enclosed with a board fence two 

boards high, each board fifteen feet long, and the whole 

number of boards enclosing the three lots is equal to the 
whole number of acres. Required the side of each lot. 


chains. 
Q : 250,61 pentagon. 
: As} 208.4 hexagon. 
172,44 heptagon. 


103. An eagle conscious of superior might, | 
Straight up through boundless ether winged his flight, 
When looking downwards through the wide expanse, 
One-third the earth’s curve surface caught his glance. 
He now desceuds:and asks the nations all, 
The heighth he soared from the terrestrial ball ? 
Would likewise know how far his piercing si. ght 
Extended from the summit of his flight ? 
Ans, To the heighth of the earth’s diameter. 
104. The area of a certain piece of land contained be- 
tween three equal circles, whose peripheries just touch 
each other, is 2 acres. What is the diameter. of that circle 
which will just enclose the three? Ans.. 47,98646 chains. 
105. A and B purchased a valuable farm, containing 
900 acres of land, at the rate of $2. per acre, which they 
paid equally between them; but on dividing the same, A 
got that part of the farm which contained the best im- 
provements, and agreed, to pay 45 cents per acre more 
than B. How many acres had each. ? 


A’s share 400 acres. 
Ans. | 3), “ 500 « 


222 QUESTIONS FOR EXERCISE. 


106. Says A to B and C, give mei, 1, 7, and 4 of 
your money,‘ and’ I shall have $50 more a I red at 
present. Says B to’ A ‘and C, give me J, 4, and ;5 of 
your money, and I shall have $50 — hg I Kare mH 
present. Says Cto A and B, give me 4; 4, 4, and 35 


and ‘my money will be doubled. How much had each . 


ea tN had. $40, 67+ 
Ans , .* 36,98-+- 
C fe *"32,00- 


107. “Suppose an ox horn to be two feet anda half long, 
and nine inches in circumference at the but, and 2 higher 
and a half at the tip; the horn being hollow, 3 4+ an inch 
thick, how much wine will it. take to fill it, and how 
much gold would i it take to overlay it half an inch | thick 


(24, 6 
Ans, 231, gallons. 
af 102, 1 solid inches. 


108. Rhee: the circumference of ine circles, the 
sum of whose diameters, if increased by 4, 4, and 4 . would 
amount to 5 times the diameter of the smaller rele, as 
if the square of their sum be increased by 3, 7, and 4, 
would amourt to 90 times their diameters: Now, a ditch 
of equal breadth and depth is to be dug around the greater 
circle, yet within the limits of its circumference, to yield 
earth sufficient to form éach circle into a half globe. Re- 
quire the width and depth of the ditch. 

A wes { Diameter 28 and 20 
~ 6,61 feet. 


“109. ‘A curious man of high dives 
A garden would lay out, , 
-' When done, intending it should be 
is Eliptically about. | 
The size of it was, if I am Hoty: 
An acre full content ; 
‘Its wall or fence, when finished quite, 
In this proportion went, 


QUESTIONS FOR EXERCISE, 223 


So that the length unto the breadth 
Should most exactly be, 
Nicely curious in the width, 
As two is unto three; 
Now, this is all I do demand 
Of any, who can tell, 
What was the breadth of this same land, 
And length also as well? 
ou ot ie 653 rods breadth. 
"17, 479 transverse Dia. 


110. Suppose, three lots of land, of equal area, one of 
which is circular, the second square, and the third in thé 
form of an equilateral triangle ; each lot is enclosed with 
a four rail fence, the rails being 12 feet in length, and the 
whole number of acres in the hice lots equal 1 to the num- 
ber of rails. Required the area of each lot, and the num- 
ber of rails enclosing it. Ans. 228970,75 acres. 

111. What is the epee between six dnzen dozen, 
and half a dozen dozen ? Ans. 792, 

112. What number multiplied by 6 will make 2058 ? 

Ans, 343. 

113. A gentleman went to seaat 17 years of age; 8 
years after he had a son born, who died at the age of 35 ; 
after whom the father lived twice 20 vears. How old was 


the father at his death ? Ans 100 years. 
114. What number is that, which being multiphed by 
15 the product will be 2 2 Ans. 35. 
115. What decimal is that, which being multiplied by 
15 the product will be ,75 1 Ans. 0% 
116. What number is that, which being divided by # 
the quotient will be 211 Ans. 153. 
117, What number is that, which multiplied by 2 pro- 
duces +1 Ans, 2. 


118. A farmer carried a load of produce to market: he 
sold 780 pounds of pork, at 6 ceuts per pound; 250 
pounds of cheese, at 8 cents per pound; 154 pounds of 
butter, at 15 cents per pound. In pay he received 60 
pounds of sugar, at 10 cents per pound; 10 gallons of mo- 
lasses, at 42 cents per gallon; 3 barrel of mackerel, at 
$3.75; 4 bushels of salt, at $1,259 per bushel; and the 
balance in money. How much money did he receive ? 

Ans. 68,853 


224 QUESTIONS FOR EXERCISE. 


119. A man exchanges 760 gallons of molasses, at 374 
cents per gallon, for 663 cwt. of cheese, at $4 per cwt. ; 
how much will be the balance in his fayor ? Ans. $19, 

120. Jones bought 84 yards of cloth at $1,25 per yard; 
how much did it come to?. How many bushels of wheat, 
at $1,50 per bushel, will it take to pay for it? 

Ans. 70 bushels. 

121. Aman sold 342 pounds of beef, at 6 cents per lb., 
and received his pay in molasses, 374 cents per gallon, 
how many gallons did he receive ? ‘Ans. 54 ‘72 galls. 

122. A man exchanged 70 bushels of rye, at 92 cents 
per bushel, for 40 bushels of wheat at $1,375 per bushel, 
and received the balance in oats at 40 cents per bushel; 
how many bushels of oats did he receive ? 

Ans. 234 bushels, 

123. How many bushels of potatoes, 1s. 6d. per bushel, 
must be given for 32 bushels of barley, 2s. 6d. per bushel 4 

7 Ans. 53% bushels. 

_ 124. How much salt, at $1,50 per bushel, must be given 
in exchange for 15 bushels of oats, at 2s. 3d. per bushel ? 
Ans. 32 bushels, 

125. How much wine, at $2,75 per gallon, must be 

given in exchange for 40 yards of cloth, at 7s. 6d. per yard ? 

Ans. 18,7, gallons. 

126. A had 41 cwt. of hops, at 30s, per cwt., for which 

B gave him £20 in money and the rest in prunes, at 5d. 
per pound ; how many prunes did A receive. — 

Ans. 17 cwt. 3 qrs. 4 lbs. 

127. A has linen cloth worth 30 cents per yard; but, 
in barter he will have 35 cents per yard; B has broad: 
cloth worth $3,75 ready money; at what price ought the 
broadcloth to be rated in bartering with A ? 

Ans. $4,374. 

128. If cloth, worth 2s. per yard, cash, be rated in bar- 
ter at 2s. 6d., how should wheat, worth 8s. cash, be rated 
in exchange for the cloth ? Ans. 10 shillings. 

129. If 4 bushels of corn cost $2, what is it per bushel ? 

Ans. 50 cts. ~ 

130. If 9 bushels of wheat cost $13,50, what is it per 
bushel ? * Ans. $1,50. 

131. If 40 sheep cost $80, what is that per head ? 

Ans. $2,00. 


QUESTIONS FOR EXERCISE, 225 


132. If 3 bushels of oats cost 7s. 6d., how much are 
they per bushel ? Ans. 2s. 6d. 
133. If 22 yards of broadcloth cost £21 9s., what is the 
price per yard ?- Ans. 19s. 6d. 
134. At 50 cents per bushel, how much corn can be 
bought for $2,002 Ans. 4 bushels. 
135. A man, having $100, would lay it out in sheep, at 
$2,50 per head, how many can he buy ? : 
Ans. 40 sheep. 
136. If 20 cows cost $300, whatis the price of 15 cows ? 
Ans. $225. 
137. If 7 men consume 24 pounds of meat in one week, 
how much would 10 men-consume in the same time 4 
Ans. 342 Ibs. 
138. If I pay $6 for the use of $100, how much must I 
pay for the use of $75. Ans. $4,50. 
139. What premium must I pay for the insurance of 
my house against loss by fire, at the rate of $ per cent, if 
my house be valued at $2475? Ans. $12,374. 
140. What will be the insurance, per annum, of a store 
and contents, valued at $9876,40 at 1iper cent ? 
7 ome Ans. $148,146. 
141. What commission must I receive for selling $478 
worth of books, at 8 per cent ? Ans. $38,24. 
142. The births in a certain town were 475, and the 
proportion 13 boys to 12 girls, what was the number of 
each ! Ans. 247 boys, 228 girls. 
143. How many yards of carpet, yard wide, will cover 
a floor 25 feet long and 18 feet wide ! Ans. 50 yards. 
144. How many trees, 4 feet apart every way, may 
grow in a nursery of one acre of ground ? 
Ans. 2722 trees. 
145. If a ship of 350 tons, chartered at 3s. a ton per 
month, deliver a cargo of 600 tons, what is the real rate 
per ton % Ans. 1s. 9d. 
146. A farmer raised 43 tons 11 cwt. 3 qrs. 14 lbs. of car- 
rots on 1 acre, 1 rood, and 25 perches. What was it 
per acre ? Ans. 31 tons, 
147. There are two numbers in proportion as 3 to 11, 
the greater-is 3267, what is the sum of both 4 | 
Ans. 4158. 
19 


226 QUESTIONS FOR EXERCISE. 


148, The Chinese wallissaid to be 1200 miles in length, 
averaging 18 feet high, and 18 aah thick, how many solid 
fathoms does it contain ? ‘Ans. 9,504,000 fathoms. 
149. Suppose a he ae. $1 103 per share, cost 
$25472,50, and that 82 of it sold for $5725, was there a 


230 
gain or loss by the sale, and how much ? 


Ans. $1141,50 iba8. 
150. A pile of wood 84 foet 6 inches long, 22 feet 7 
inches high, 23 feet 10 inches wide, is sold at $3,26 per 
cord, what is the amount? Ans. $1158,32. 
151. Required the cost of a lot of land 62 feet 11% 
inches long, and 27 feet 34 inches wide, at $1.80 per 
square foot? Ans: $3093,>6. 
152.-A of Boston has in his hands $500 due to M of 
Baltimore, for net proceeds of his cotton, this he remits 
to M per bill on D in his favor when bills on’ Baltimore 
are 2 percent. discount. Require the amount of the bill ? 
Ans. $510,20. 
153. Suppose $984, 3724 was paid in New Orleans fora 
bill on New York, when vohie advance’ was 5 per cent. ; 
what was the bill drawn for? Ans. $937,50. 
154. If 33865 feet of land sold in Boston in 1824 for 
$403840,124, how much is it per foot, and what would 
be the rate per acre in federal money, and:also it sterling 
money ? Ans. $11,923 per foot. 
$519,458 per acre. 
Equal to £116,876, 18, 6 sterling. 


155. How much land, at $250 per acre, must be given 
in exchange for 360 acres, at$3,75 per acre? | 
Ans. 540 acres. 
156. A merchant bought a quantity of goods for $734, 
and sold them so as to gain 21 per cent. ; for how much did 
he sell his goods? Ans, $888,14. 
157. A merchant bought a quantity of goods at Boston 
for $500, and paid $43 “for their transportation, he sold 
them so as, to gain 24 per cent. on the whole cost; for 
how much did te sell them ? Ans. $673,32. 
[58. Bought a quantity of books for $64, but for cash 
a discount of 12 per cent was made; what did the books 
cost 2 Ans. $56,32. 
159. Bought a book, the price of which was marked 


QUESTIONS FOR EXERCISE. 227 


$4,50, but for cash the bookseller will sell it at 332 ver 
cent. discount ; what is the cash price ? » Ans. $3,00. 
160. A merchant: bought a cask of molasses, containing 
120 gallons for $42; for how much must he sell it to gain 
15 per cent; how Se per gallon ? 
Ans. to the last $,403. 
161. A. merchant bought a cask of sugar containing 
740 pounds, fer $59.20 ; how must he sell per pound to 


gain 25 per cent! Ans. $,10 cts. 

162. What is the interest, at 6 per cent:, of 71,02. for 

17 months 12 days] Ans: $6; 178--. 

163. What is the interest of $487,003 for 15 mouths, at 

6 per cent! Ans $13,83-+. 
[64. What is the interest of €8,59 for 7 mouths ! 

Ans. $,2974. 


165. What is the interest of $1000 for 5 days? 
Ans. #, S34 cts. 
166. Whatis the interest for 50 cents for 10 years 4 


Ans,'30 cts, 

167. What is the interest of $84,25 for 15 months and 7 
days, at 7 per cent./ Aus. $7.48,5.. 
168. What is the interest of $154,01 for 2 years, 4 
months and 3 days, at 5 per cent ? Ans. $18 032. 


169. What sum, put at interest at 6 per cent., will, in 2 
years and 6 months, amount to $1501 
Ans. $130,434-++. 
~170: Towe a man $475,50, to be paid in 16 months, 
without interest; what is the ‘present worth of that debt, 
the use of the money being worth 6 per cent ? 
Ans. $440, 277--, 
171. Whatis the present worth of $1000 payable in 4 
— and 2 months, discounting at the rate of 6 per cent? 
Ans. $800. 
172. A merchant bought articles to the amount of #500, 
and sold them for $575, what did he gain per cent. ? 
Ans, 15 per cent! 
A173, A meréhant bought cloth at $3,50 per yard, and 
sold it at aS 25 per yard, how much did he gain per cent ? 
Ans. 212 per cent, 
174. A man bought a cask of wine, containing 126 gal- 
lons, for $283,50, and sold it out at the rate of "$2,75 per 


*~ 


228 QUESTIONS FOR EXERCISE. 


gallon ; how much did he gain on the whole; how much 
per gallon and how much per cent? 

Ans. whole gain $63; per gallon $, 50, 

‘ which is 222. per cent. 

175. If $100 gain $6, in 12 months, in, what time will it 


gain $14? Ans. 24 months. 
176. In what time will $54,50, at 6 per cent., gain 
$2,18 4 ‘Ans. 8:months. 


177. 20 men built a certain bridge in 60 days, but. it 
being carried away in a freshet, it is required how many 


men can rebuild it in 50 days ? i Ans. 24 men. 
178. Ifa field will feed 7 horses 8 weeks, how long 
will it feed 28 horses ? Ans. 2 weeks. 


179. Ifa field, 20 rods in length, must be 8 rods in 
width to contain an acre, how much in width must be a 
field, 16 rods in length to contain the same? 

Ans. 10. rods. 

180. fl purchase for a cloak 12 yards of plaid 3 of a 
yard wide, how much hocsing. 14 yards wide, must I 


have to line it? Ans. 5 yards. 
181. Ifa man earn $75in 5 months, how long must he 
work to earn $460? Ans. 302 months. 


182. A owes B $450, but A not being worth. so much 
money, B. agrees to take 75 cents on the dollar; what 
sum must B receive for the debt? Ans. $405. 

183. A cistern, whose capacity is 400 gallons, is sup- 
plied by a pipe which lets in 7 gallons in 5 minutes, but 
there is a leak in the bottom of the cistern which lets out 
2 gallons i in 6 minutes, supposing the cistern empty, in 
what time would it be ‘filled 3 

Ans. 6 hours 15 minutes. 

184. A ship has a leak which will fill it so as to make it 
sink in 10 hours, it has also a pump which will clear it in 
15 hours, now if they begin to pump when it begins to 
leak, in what time will it sink 2 Ans. 30 hours. 

185. A cistern is supplied by a pipe which will fill it in 
40 minutes ; how many pipes of the same bigness will fill 
it in 5 minutes 2 Ans. 8. 

186. Suppose I lend a friend $500 for 4 months, he 
promised to do me a like favor; some time afterward I 
have need of $300, how long may I keep it to balance the 
former favor ? Ans. 63 months 


\ 


QUESTIONS FOR EXERCISE. 229 


187. Suppose 800 soldiers were in a garrisen with pro- 
visions sufficient for 2 months; how many soldiers must 
depart, that the provisions may serve them 5 months? 

Ans. 480. 

188. If my horse and saddle are worth $84, and my 
horse be worth 6 times as much as my saddle, pray what 
is the value of my horse.? Ans. $72. 

189. -Bought 45 barrels of beef at $3,50 per. barrel, 
among which are 16 barrels, whereof 4 are worth no 
more than 3 of the others; how much must I pay! 

Ans. $143,50. 

190. Bought 126 gallons of rum for $110, how much 
water must be added to reduce the first cost to 75 cents 
per gallon? Ans. 202 gallons. 

191. A thief having 24 miles start of the officer, holds 
his way at the rate of 6 miles an hour; the officer press- 
ing on after him at the rate of 8 miles an hour; how long 
before he will overtake the thief? Ans. 12 hours. 

192. In an orchard of fruit trees, 4 of them bear ap- 
ples, + pears, plums, 60 of them peaches, and 40 cher- 
ries; how many trees does the orchard contain 4 

Ans. 1200 trees. 

The above example and others are usually wrought by 
the rule called Position, but they are more easily solved 
on general principles. 

193. A and B commenced business with equal sums of 
money; A gained a sum equal to + of his stock, but B 
lost $200, and then had only half as much as it, what was 
the original stock of each ? - Ans. $900. 

194. By Ierguson’s tables of specific gravities a cubic 
inch of pump water weighs 9,26 dramas, and it is found on 
trial that a gallon of 231 cubic inches of cider-weighs 10 
ounces and 10 drams more than water; what then should 
the liquor in a barrel of 314 gallons weigh ? 

: Ans. 284 pounds. 

If the gallon be taken at 9 pounds it will answer for 
common purposes. 

_ 195. There are 7 chests of drawers, in each of which 
there are-18 drawers, and in each of these there are 6 
divisions, in each of which is £16 6s, 8d.; how much 
money is there in the whole ? Ans, £12348, 


e. 


230 QUESTIONS FOR EXERCISE, 


196. Bought a piece of cloth for $50, at 75 cents per 
ey and sold + of it at 10 per cent. gain, the remainder 
at 15 per cent. foss ; what was the loss on the whole piece ? 

Ans. $1,25. 

.197. A hare starts 12 rods before a hound, but-is not 

perceived by him till'she has been up 1} minutes; she 

scuds away at the rate of 36 rods'a’ minute, and the dog 

on view makes after, at the rate of 40 rods a minute ; ; ow 

long willthe course hold; and what distance will the dog 
run ? Ans. 141 minutes, he will run 570 rods’ 

198. A person who was possessed of 2 of a vessel, sold 
3 of his interest for €375 ; what was ae ship worth at that 
rate ? Ans. £1500. 

199. A man was hired for a term of 50 days: on ¢on- 
ditions, that for every day he worked he should receive 75 
cents, and for’ every day he was idle he should-pay 25 
cents for his board; at the expiration of the time, he ree 
ceived $27,50, how many days did he work ? © 

Ans. 49 days. 

200. Band C purchased 1200 acres of land ‘at one dol- 
lar per acre, each paying $600. Some time after, Con 
Viewing it, offers to take a certain square piece at $1,75 
per acre to the amount of his advance, to which B con- 
sents; how many acres will each have, what is the length 
of each side of C’s lot, and what does B’s part cost him 
per acre ? 

Ans. C has 342 acres, 3 roods, 172 rods. 
B ‘“ 857 re 6) 73 208 He 
B’s land is 70 cents per nates 
Side of C’s =234 rods 3 ft. 62 in. 


201. Bought 36° pipes of wine for $4536; how must I 
sell it a pipe to save one for my own use, ana sel] the rest 
for what the whole cost ? Ans. $129,60 cts. 


202. Two drovers meeting on their way, 
And thus they said—‘’ Tis true, 
If half your flock you give to me 
I'll have just eighty-two, 
“« Nay, friend,’’ the other soon replied, 
‘Add but athird to mine . 
Of your best sheep —then I shall have 
One hundred tweuty-nine.”’ 


& 


he 


QUESTIONS FOR EXERCISE. 231 


His answer was exactly true, 
~ Noscholar will impeach ; 
Then by your knowledge show to me 
‘How many sheep had each? 
4 Ans. A Wha at. 
Bh eT 22. 


203. The hour and minute hands of a watch are exactly 
together at 12 o’clock ; when are they next together! 

Ans. | hour 6 m,. 277, sec. 

204. If ? of 3 of 4 ofaship be worth 2 of 4 of 12 of the 

cargo, valued at £1000, what did the ship oT cargo cost ! 

Ans. £1837 12s. 15d. 

205. A and B have the same income; A. saves 4 of 

his, but B by spending $30 a year more than A, at the 

end of 8 years finds himself $40 in debt, what is their in- 

come, and what does each spend a year 4 

Ans. Income $200. 

A spends $175. 

Dao pete. 


206. Three parcels of beef, of 60 barrels each, were 
received at Baltimore from Boston, marked, viz: W.M. 
Y. The lot marked W. was found to be 50 per cent, bet- 
ter than the others. If the whole sold together at $10 per 
barrel, how must the sales be Pte between the own- 
ers Of ‘the beef ? 


Ans. Y’s 60 bbls. at $8,574 — $514,282, 


, M's 60 g57L— 514,984, 
W's 60 “12,858 771,428. 
180 $10,00 $1800,00 


207. Just 16 yards of German serge 
For 90 dimes had I; 

How many yards of that same cloth 
Will 14 eagles buy % 
| Ans. 248 yards, 3 qrs. 22 n 
208. A gentleman divided lis fortune among his 3 sons, 
giving A 8 dollars as often as B 5, and C but 3 as often as 
B 7, yet C’s share amounted to $1200; what was the 
father’s estate ? ~ Ans. $8480. 


232 QUESTIONS FOR EXERCISE. 


209. There is an island 20 miles in circumference, and 
3 men started together to travel the same way about it ; 
A goes 2 miles per hour, B 4 miles per hour, and-C 6 
miles per hour; in what time will they come together 
again ¢ Ans. 10 hours. 

210. A man and his wife can drink a cask of beer in 12 
days, but when the man is from home it lasted the woman 
30 days, how many days would the man be in drinking it 
alone ? Ans. 20 days. 

211. The roof of a building with perpendicular front 
makes with the horizon an angle of 45 degrees, a leaden 
ball rolled from the apex thereof strikes the horizontal 
plane below 40 feet from the base of the front, but when 
rolled from the centre of the roof it strikes only 30 feet 
from the base. Required the heighth of the front and 
length of the roof ? se 
Ans, 120 (front) feet 14,142 feet. 

212. An iron ball rests upon three contiguous balls of 
the same diameter, fixed upon a horizontal plane, but if 
the i1on ball be exchanged for one of the same material 
and double its diameter, the whole pressure upon the 
three balls will be increased 40 pounds. Required the 
diameter of each ball ? . Ans, 6,736 inches. 

213. Two men, A and B, laid hold on a parcel of ap- 
ples, and after devouring them discovered that A had 
eaten 15 apples more than B, A would have been 36 
minutes eating B’s apples, and B would have required 49 
minutes to have eaten A’s. Required the time they were 
eating and the number of apples, and how many were de- 
voured by each ? 


105 apples dev gag by A. 


Time 42 minutes. 
Ans 
90 sx 5 Be 


214. Suppose a square yard containing 36 perches was 
enclosed, at one of the corner posts a horse was fastened 
by a line just long enough to reach round the enclosure, 
Required what quantity of land he could graze? 

Ans. 1745,6 rods. 

215. A and B purchased a farm containing 450 acres at 
$6 per acre, for which they paid equally, A getting the 


QUESTIONS FOR EXERCISE, 233 


part with the improvements agreed to pay $1 more per 
acre than B... How much land had each, and at what 
price per acre ! 

ae f A’s206--acres at $6,54-++. 

"Bis 243-- tt ot Oyba--. 

216. Borrowed a sum of money at 8 per cent. simple 

interest, and lent it out again at 5 per cent., compound 
interest. When shall 1 gain the amount borrowed ? 

Ans. 304 years. 


- 217. Three men, A, B and C, sent a ship te Cuba, with 
indigo, to the amount of $473344, A bought 250 cwt. 2° 
qr. 22 lbs., at $84 per cwt., B paid $70 per cwt. for his ; 
but meeting with a storm at sea, they lost part overboard, 
A’s proportional part cast overboard was equal to the ,4, 
part of the whole cargo, and 33% times the whole quantity 
cast overboard was equal to 34 times the whole cargo of A 
and B. When they came to land, A sold his remaining part 
for $126 per cwt., and found himself the loser of 10 per 
cent., besides charges B advanced the remaining part of 
his commodity 20 percwt., and C gained $7 per cwt. by 
the quantity he sold. What did each person lose by this 
voyage, the charges whereof amounted to $15750 ? 
A’s loss $2497,50. 
an.) B’s -** 90142,50. 
C’s  * 47231,25. 


218. A, in a scuffle, seized on 2 of a parcel of sugar 
plums, Bcatched 2 of it out of his hands, and C laid hold 
on ;3; more; D ran off with all A had left, except + which 
E afterwards secured slyly for himself; then A and C 
jointly set upon B, who, in the conflict shed § he had, 
which were equally picked up by D and E, who lay perdue, 
B then kicked down C’s hat, and to work they all went 
anew for what it contained; of which A got }, B 4, D2, 
C and E equal shares of what was left of that stock; D 
then struck ?of what A and B last acquired, out of. their 
hands; they with difficulty recovered % of it in equal 
shares again, but the other three carried off + a piece of the. 
same. Upon this they called a truce, and agreed that the 


234 QUESTIONS FOR EXERCISB. 


+ of the whole left by A.at first should be equally divided 
among them. How much of the:prize, aiter this distribu- 
tion, remained with each of the competitors ? 


’ ¢ 2863 

{ A's partaeeso 

‘ 9) 

eee Psteyarivle: ~~ 
ns. Ss SARRNHN? 
Digi GOED 
26880°* 

| Hs “ Aoee 


219. A lad having got 4000 nuts, in his return home 
was met by mad Tom, who took from him 3 of 72 of his 
whole stock; raving Ned lights on him afterwards, and 
forced 2 of 2 of the remainder from him, unluckily, posi- 
tive Jack found him, and required 7 of 44 of what he 
had left ; Smiling Dolly was, by promise, to have } of 4 
of what nuts he brought home. _ How. many then had the 
boy left % Amelie dis. 

220. A and B havea certain number of dollars, says A 
to B, multiply the square root, of your dollars by mine, and 
the number will be $180, and says B to A, multiply the 
square root of your dollars by mine, and it will.be $150. 
Required the number of dollars of each ? | 

No A’s $36. 
“B25. 

221. Laid out ina lot of muslin £500, but upon exumi- 
nation 3 parts in 9. proved to be damaged, so that I could 
make but 5s. per yard of it, and. by so doing find I lost 
£50, at what rate per ell must I sell the undamaged part 
sothat I may clear £50 by the whole 4 
Ans, 11s, 72d. 

222. A and B together can perform a piece of work in 
8 days, A and C together in 9 days, and Band C in 10 
days. How many days will it take each person to per- 
form the same work alone 4 

4 
9 
Ans.<4.B “ 1733 


LC 6c 2335 a4 


( A in 1434 days. 
Ce 


223. I would plant 10 acres of hop ground, which must 
be done either in the square order, asthe numbe: 4 stands 
on the dice, or in the quincunx order, as the number 5; 


QUESTIONS FOR EXERCISE. 235 


the three nearest binds, in both cases’ must be set lineally 
just 6 feet asunder.. How many plants more’will be re- 
quired for the last order than the first, admitting the form 
of the plot to lay the most advantageous for the ; plantation 
in either case ? Ans. 1872 more. 
224, A water tub holds 147 gallons, the pipe usually 
brings in 14 gallons in 9 minutes, the tap discharges.it at 
a medium of 40 gallons in 31 minutes. Now these to be 
carelessly left open, and the water to be turned on at 2 
o’clock in the morning, a servant at 5 finding the water 
running, shuts the tap, and is solicitous to know in what 
time the tub will be filled after this accident, in case the 
water continues to flow from the main ? 
_Ans. 6 0’clock, 3 minutes 48225 sec. 
225. A, Bye and-D agree to build a house ; A, B and 
C can build it in 69 days, B, C and Din 87 days, C, D 
and “Ain 100 days, and D, ‘A and B in 120 Hoyas in what 
time would a = build it? 
Ans. 672 days (nearly.) 
226, In se time would each person build the above 
mentioned house alone? | 


(AE: 304-4 days. 


'B 209+ « 
ADS. 9.6) 4664-5166 
lp 3563+) « 


227. A°’merchant in England can draw directly for 1000 
piasters in Leghorn at 6 pence sterling per piaster, but he 
chooses to remit the same to Cadiz at 19 piasters for 7000 
maravedies, thence to Amsterdam at 189 pence Flemish 
for 680 maravedies, and thence to Liverpool at9 pence 
Flemish for 5 pence sterling. What is gained by this cir- 
cular remittance, and what ‘is the value of a plaster to him 4 

Ans. gain £28 14s., value 56-3,55. 

298. A merchant in New York orders £500 sterling due 
him at London at 54 pence sterling per $ to be sent by 
the following circuit to Hamburg it 15 marks banco per 
£ sterling, thence to Copenhagen at 100 marks banco for 
33 rix $s, thence to Bordeau at one rix $ for 6 francs, 
thence to Lisbon at 125 francs for 18 milrees, and thence 
to New York at $14 per milree. Did he gain or lose by 


236 QUESTIONS FOR EXERCISE. 


this circular remittance, and what was the arbitrated welts 
of a $ by this remittance ? 
Ans. he gained, value of a $, 69 penne. 

229. A gentleman sold a watch for $24, and gained 
as much per cent. as the watch costhim. Required the 
cost of the watch 1? Ans. $20. 

230. A. stationer sold quills at $1,50 a thousand, 4 of 
which was profit. When they became scarce he raised 
the price to $2,25 per thousand. How much per cent. 
did he gain by the latter price 2 


150 ©, —225 | 100 

4 374 Gi 
—-— 295—2 
1,124 ih pin ate 


100 per cent. 


231. A fox starts 80 yards before.a hound, and is not 
perceived by him till he has been up 45 seconds, he scuds 
away at the rate of 9 miles an hour, and the hound pur- © 
sues after him at the rate of 18 miles per hour. In how 
many seconds will the hound overtake the fox, and how 
far will each have run ? 

The fox has 80 yards the start, we must see according 
to the conditions of the question, how many yards he has 
run in 45 seconds, thus : 


—4 —20 —60 | 45— 9 


—2 60 
9— 3— 
| 1760— 22 
198 Ans. 


80 the start. 


Advance before the hound starts 278 


Let us now ascertain in how many seconds the hound 
will gain 278 yards and overtake the fox 1 


11 —22 —44 —1760 | 278— 139 
—3 —9 | 60— 2— 
60— 20— 5 


eee 


11 | 695=63,4, seconds, Ans. 


QUESTIONS FOR EXERCISE. ° 


237 


Finally, we have to ascertain how many yards the hound 


has run in 637, seconds. 


—11 | 695— 139 
—5 —20 —60 | 18— 3— 
—60 | 1760— 16— 4 © 
556 Ans. 
556 yards the hound has run. 
The fox had 80 ‘. the start. 


476 yards the fox has run. 


232. A wall is to be built 80 feet long, 45 feet high, and 
23 feet thick. How many bricks will it take, if each brick 
is 45 inches in length, 4 inches broad, and 2 thick ? 


80— 20 
45— 
—2 | 5 
- 1728 
Oey a Te 
—4 
<5 {2 


| 345600 bricks, Ans. 


233. A wall of 80 feet in length, 45 feet high and 24 
feet thick, has been built with 345600 bricks, of 4 inches 
broad and 24 inches thick. What was the length of each 
brick in inches? 3 


DENY) Li BD fla 
—5 | 80— 2— 
12— 
—345600 | 45— 9 
—2880 12— 
—240 —2 | 5—. 


) 10 220 je 


2 | 9=44 inches, Ans. 


234. Suppose a wheel of 45 feet diameter, turns 10 
times round in one minute, having a similar wheel on the 
other end of its shaft, which runs in a wheel of 6 feet 


238 QUESTIONS FOR: EXERCISE. 


diameter, on whose shaft.is a wheel of 30. feet ‘diameter, 
which runs in-a wheel of 8 feet diameter, which has a 
shaft with a wheel of 15. feet diameter, which runs in .a ~ 
wheel of 5 feet diameter, which-also has a shaft holding a 
wheel of 12 feet diameter, which runs into one of 3. feet 
diameter. How often will the last wheel, of 3 feet diame- 
ter, turn round in one minute ? 


ab Apo 
—2 —4 —S§ | 30— 15 
45 oe 
10s oe 


ee er ee 


| 3375 times per minute Ans. 


235. A, on preparing for a voyage to Calcutta, pur- 
chased of G specie dollars to be paid in 18 months with 
interest. Supposing the premium on the dollars to be 3 
per cent., and that G would have a compensation of 5 per 
cent. per annum for the use of his money, to be inserted 
in the note, which was given for $22145; I would know 
the sum purchased ? Ans. $20060. 

236. Two merchants, B ea C,; trade together; B ad- 
vances $5000, and at the end of 4 months, being pressed 
for money to answer a demand, he takes out a certain 
sum, leaving the remainder to continue 8 months ; C ad- 
vances $2500, and at the end of 5 months he finds it neces- 
sary to put in $3000 more, and continues the whole 7 
months longer, when they close their business, and B finds 
he has gained $10662 and C $13331, Iwould know how 
much B took out at the end of 4 months ? 

Ans. $2400. 

237. G bought and sold for cash the following lots of 
flour, viz.; Jan. lst he bought 50 bbls. at $5,745 per bbl.; 
on 15th, 20 bbls. at $5,60; on 16th he sold 65 bbls. 
at $6,25; on 17th he bought 10 bbls. at $6,75; June 5th 
he bought 16 bbls..at 5,50; and on the 19th, 19 bbls. 
at $6, 80; on the Sth August he bought 30 bbls. at. $6,50, 
and on the 25th he sold 68 bbls. at $6,60; Sept. 12th. he 
bought 43 bbls. at $5,80; on 15th he sold 10 bbls. at $6, 
and, on 18th, 30 bbls. at $5, 60; on 5th Oct: he bought 15 — 
bbls at $6, and on 24th he sold 20 bbls. at 86, 124. How 


QUESTIONS FOR EXERCISE, 239 


many. bards has. ae on Sead) and what is his gain or loss, 
estimating what remains at $6, 25: per barrel? - 
"gain $49, 45. 


238. Two Sarptiiters, Bae nil B, who have each an ap- 
prentice, engage to finish a piece of work for $630. By 
agreement between them, A’s apprentice is to be allowed 
623 cents per day, and Bs 100 cents. When the work 
was finished, it appeared that A worked 120 days, and his 
apprentice 100; B worked 96, and his apprentice 1354 
days. Supposing that, while doing the work, they re- 
ceive each $210, what is each person’s share of the remain-. 
ing payment, on stating their accounts | i 
Ans. “A $92.50 due. 

wAa BLVT,OU. 


239. “Saiaee and John have lived together 8 years in 
John’ s house, the rent of which is stated at $50 per an- 
num.  James’s bill for supplies is $1646,46, and John’s 
bill $497,24, and he hasJames’s note for $460,80, without 
interest. Required, the balance on stating their accounts, 
and in whose favor? Ans. $236,19 due Jonn. 


240. On Ist of May B of Boston had of H of Lowell 10 
bbls. of flour at $6,75 cts. per bbl., and paid him in part 
$25 in cash. On 15th he had of H 31 gallons of molasses 
at 30 cts. and a bbl. at 83 cts,;. 19th he delivered to H 30 
qtls. of fish at $2,50 cts., and took 20 yards of baise at 50 
cts; June 3d, B had 250 Ibs. of coffee at 24 cts., and 10 
Ibs. of chocolate. ‘at 25 cts.; July 27, B brought to H 4 
bbls. of oil at $10, and 31st he sent to H 4 bbis. Salmon, 
at $10,50 cts., when H. paid his order to J. M. for $12,50 
ets., and delivered per his order to D, L. 903 Ibs. of 
sugar at 7 cts.; Sept. 6th, A paid B’s note to G for. cor- 
dage, on ee H was gitioree. viz., for principal, $65,94 
cts., $1,87 cts. for interest. On 10th B brought to him 
5854 feet of boards, at $11,50 cts. per thousand, and 10 bar- 
rels of No. 1 hovel at $5; asettlement was then made, 
and. he was furnished with ae account, and the balance 
paid in cash. What was the amount, and in whose favor ? 
: Ans. $5,67 cts. in favor of B. 


241..A, B, and C agreed on an entertainment, to which 
some friends were invited; A and B supplied the provi- 


Ans 


240 QUESTIONS FOR EXERCISE. 


sions, &c., in 8 baskets of equal cost, five of which were 
supplied by A and three by B. When the entertainment 
was finished, C laid down $12,64 cts. for his part, which 
was to be shared by A and B; but disagreeing in the 
division of it, they referred it to D, who awarded 16 each 
his part, and provide the justness of his decision by stating 
it in an account. Required, the amount awarded to each? 

. ae A $11,06 ce 

1 ADs Ab pee. 


242. A person, failing 1 in trade, owed to A $100, B $200, 
C $400, to D $350; and his property. consisted of 

334 yards Br oadeloth, worth $5,75 cts. per yard, 

512 ff | Cassimere;— 2,46 


1362 «Linen, i 86 A 

2293. * Wiknnch f Ris) ¢ 

58. lbs. Tea, as 1,20 per lb. 

954 EPS i Sugar, 66 ,09 66. 
3. bbls. Flour, ss 575 per bbl., 


which was assigned for benefit of his creditors. The com- 
mission on sale of the goods, at the appraised value, was 
at 21 per cent., and the assignee’s bill $43,50 cts.; on ex- 
hibiting their statement to the creditors and paying the 
ainsi dees: how much was there paid to each, and how 
much did Pe pay on the dollar ? 
( To A $57,14 cts. } 
Ana “ B 114,29 cts. (574 cts.on 
"\ « © 228,57 cts. + the dollar. 
| « D 200,00 ets. § 


ERRATA. 


Page 21, Ist line, for 17 read 19. ° 


25, 6th line from the bottom, for 36,000 read 36,900. 


- 56, 23d line from the top, read X instead of ++. 


57, 7th line from the bottom, read X instead of +. 


59, 11th and 12th lines from the top, read X all five of 


them. These are all signs of meutnphieaven, instead 
of addition. 


63, 4th line from the top, for 49 read 49—-. 
‘64, for + read X. 


67, 8th line, Ans., for 2 read 14, 

67, 26th Example, for 1? ® read 10. 

66, the 27th Example wn ibis page belongs at the bottom 
of page 67. 

71, 3d line from the bottom, for 3,read al 

72, 4th line from the bottom, for 3 read 3—-. 

73, 5th Example, read + , 

74, 3d line from the top, for 2— 6 read 0—. 

60, 2d line from the top, read 44 yards cost. 


64, 2d line from the top, read & of the eargo. 


89, 123d Example, read 4, 

105, 3d line of 2d Example, for 96— 50— read 9650—. 

106, 5th line of 6th Example, read 4-9. 

115, Ist Solution, read X & X, and 840 instead of 746. 

126, 2d line from tup, read Tare and Tret are. 

1538, 3d Ex. 5th lihe, read 1210— instead of 12,1— 0—. 

201, 11th line from the bottom, instead of 3 read-8. 

207, 3ist Ex. 9th line, instead of 26.876, read 26,776. 

212, 10th line from the top, instead of 8, * O1, 335, read 
at $3, 01,335. 

233, 217th Example 2d line, for 2.qr. read 1 gr. 


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ST Te 
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A ay 
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